A sphere () is a

^{3}.

circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

. This second part also holds for the

spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

with held constant.
A sphere of any radius centered at zero is an

geometrical
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

object that is a three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...

analogue to a two-dimensional circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

. A sphere is the set of points
A railroad switch (), turnout, or ''set ofpoints () is a mechanical installation enabling railway trains to be guided from one track to another, such as at a railway junction or where a spur or siding branches off.
The most common ty ...

that are all at the same distance from a given point in three-dimensional space.. That given point is the centre
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentricity ...

of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...

.
The sphere is a fundamental object in many fields of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble
Bubble, Bubbles or The Bubble may refer to:
Common uses
* Bubble (physics), a globule of one substance in another, usually gas in a liquid
** Soap bubble
* Economic bubble, a situation where asset prices are much higher than underlying funda ...

s such as soap bubble
A soap bubble is an extremely thin film of soap or detergent and water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact wi ...

s take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography
Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and ...

, and the celestial sphere
In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, ...

is an important concept in astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

. Manufactured items including pressure vessels
A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure.
Construction methods and materials may be chosen to suit the pressure application, and will depend on the size o ...

and most curved mirror
A curved mirror is a mirror with a curved reflecting surface. The surface may be either ''convex'' (bulging outward) or ''concave'' (recessed inward). Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are ...

s and lens
A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...

es are based on spheres. Spheres roll
Roll or Rolls may refer to:
Movement about the longitudinal axis
* Roll angle (or roll rotation), one of the 3 angular degrees of freedom of any stiff body (for example a vehicle), describing motion about the longitudinal axis
** Roll (aviation), ...

smoothly in any direction, so most ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

s used in sports and toys are spherical, as are ball bearings
A ball bearing is a type of rolling-element bearing that uses balls to maintain the separation between the bearing races.
The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. It achieves this ...

.
Basic terminology

As mentioned earlier is the sphere's radius; any line from the center to a point on the sphere is also called a radius. If a radius is extended through the center to the opposite side of the sphere, it creates adiameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...

. Like the radius, the length of a diameter is also called the diameter, and denoted . Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius, =. Two points on the sphere connected by a diameter are antipodal point
In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true d ...

s of each other.
A unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...

is a sphere with unit radius (=1). For convenience, spheres are often taken to have their center at the origin of the coordinate system, and spheres in this article have their center at the origin unless a center is mentioned.
A ''great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...

'' on the sphere has the same center and radius as the sphere, and divides it into two equal ''hemispheres''.
Although the Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere.
If a particular point on a sphere is (arbitrarily) designated as its ''north pole'', its antipodal point is called the ''south pole''. The great circle equidistant to each is then the ''equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...

''. Great circles through the poles are called lines of longitude
Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...

or meridians. A line connecting the two poles may be called the axis of rotation
Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...

. Small circles on the sphere that are parallel to the equator are lines of latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

. In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and ''noted'' as such, unless there is no chance of misunderstanding.
Mathematicians consider a sphere to be a two-dimensional closed surface
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...

embedded in three-dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. They draw a distinction a ''sphere'' and a ''ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

'', which is a three-dimensional manifold with boundary
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...

that includes the volume contained by the sphere. An ''open ball'' excludes the sphere itself, while a ''closed ball'' includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...

of a (closed or open) ball. The distinction between ''ball'' and ''sphere'' has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

" and "disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...

" in the plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...

is similar.
Small spheres are sometimes called spherules, e.g. in Martian spherules
Martian spherules (also known as hematite spherules, blueberries, & Martian blueberries) are small spherules (roughly spherical pebbles) that are rich in an iron oxide (grey hematite, α-Fe2O3) and are found at Meridiani Planum (a large plain on ...

.
Equations

Inanalytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineerin ...

, a sphere with center and radius is the locus
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** ''Locus Award' ...

of all points such that
:$(x\; -\; x\_0\; )^2\; +\; (y\; -\; y\_0\; )^2\; +\; (\; z\; -\; z\_0\; )^2\; =\; r^2.$
Since it can be expressed as a quadratic polynomial, a sphere is a quadric surface, a type of algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...

.
Let be real numbers with and put
:$x\_0\; =\; \backslash frac,\; \backslash quad\; y\_0\; =\; \backslash frac,\; \backslash quad\; z\_0\; =\; \backslash frac,\; \backslash quad\; \backslash rho\; =\; \backslash frac.$
Then the equation
:$f(x,y,z)\; =\; a(x^2\; +\; y^2\; +z^2)\; +\; 2(bx\; +\; cy\; +\; dz)\; +\; e\; =\; 0$
has no real points as solutions if $\backslash rho\; <\; 0$ and is called the equation of an imaginary sphere. If $\backslash rho\; =\; 0$, the only solution of $f(x,y,z)\; =\; 0$ is the point $P\_0\; =\; (x\_0,y\_0,z\_0)$ and the equation is said to be the equation of a point sphere. Finally, in the case $\backslash rho\; >\; 0$, $f(x,y,z)\; =\; 0$ is an equation of a sphere whose center is $P\_0$ and whose radius is $\backslash sqrt\; \backslash rho$.
If in the above equation is zero then is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adj ...

..
Parametric

Aparametric equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...

for the sphere with radius $r\; >\; 0$ and center $(x\_0,y\_0,z\_0)$ can be parameterized using trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

s.
:$\backslash begin\; x\; \&=\; x\_0\; +\; r\; \backslash sin\; \backslash theta\; \backslash ;\; \backslash cos\backslash varphi\; \backslash \backslash \; y\; \&=\; y\_0\; +\; r\; \backslash sin\; \backslash theta\; \backslash ;\; \backslash sin\backslash varphi\; \backslash \backslash \; z\; \&=\; z\_0\; +\; r\; \backslash cos\; \backslash theta\; \backslash ,\backslash end$
The symbols used here are the same as those used in spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

. is constant, while varies from 0 to and $\backslash varphi$ varies from 0 to 2.
Properties

Enclosed volume

In three dimensions, thevolume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

inside a sphere (that is, the volume of a ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

, but classically referred to as the volume of a sphere) is
: $V\; =\; \backslash frac\backslash pi\; r^3\; =\; \backslash frac\backslash \; d^3\; \backslash approx\; 0.5236\; \backslash cdot\; d^3$
where is the radius and is the diameter of the sphere. Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the circumscribe
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...

d cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...

of that sphere (having the height and diameter equal to the diameter of the sphere). This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying Cavalieri's principle
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:
* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...

. This formula can also be derived using integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

, i.e. disk integration
Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method mod ...

to sum the volumes of an infinite number
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to qua ...

of circular
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circula ...

disks of infinitesimally small thickness stacked side by side and centered along the -axis from to , assuming the sphere of radius is centered at the origin.
At any given , the incremental volume () equals the product of the cross-sectional area of the disk at and its thickness ():
: $\backslash delta\; V\; \backslash approx\; \backslash pi\; y^2\; \backslash cdot\; \backslash delta\; x.$
The total volume is the summation of all incremental volumes:
: $V\; \backslash approx\; \backslash sum\; \backslash pi\; y^2\; \backslash cdot\; \backslash delta\; x.$
In the limit as approaches zero, this equation becomes:
: $V\; =\; \backslash int\_^\; \backslash pi\; y^2\; dx.$
At any given , a right-angled triangle connects , and to the origin; hence, applying the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

yields:
: $y^2\; =\; r^2\; -\; x^2.$
Using this substitution gives
: $V\; =\; \backslash int\_^\; \backslash pi\; \backslash left(r^2\; -\; x^2\backslash right)dx,$
which can be evaluated to give the result
: $V\; =\; \backslash pi\; \backslash left;\; href="/html/ALL/l/^2x\_-\_\backslash frac\_\backslash right.html"\; ;"title="^2x\; -\; \backslash frac\; \backslash right">^2x\; -\; \backslash frac\; \backslash right$
An alternative formula is found using spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

, with volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV = \ ...

: $dV=r^2\backslash sin\backslash theta\backslash ,\; dr\backslash ,\; d\backslash theta\backslash ,\; d\backslash varphi$
so
: $V=\backslash int\_0^\; \backslash int\_^\; \backslash int\_0^r\; r\text{'}^2\backslash sin\backslash theta\backslash ,\; dr\text{'}\backslash ,\; d\backslash theta\backslash ,\; d\backslash varphi\; =\; 2\backslash pi\; \backslash int\_^\; \backslash int\_0^r\; r\text{'}^2\backslash sin\backslash theta\backslash ,\; dr\text{'}\backslash ,\; d\backslash theta\; =\; 4\backslash pi\; \backslash int\_0^r\; r\text{'}^2\backslash ,\; dr\text{'}\backslash \; =\backslash frac43\backslash pi\; r^3.$
For most practical purposes, the volume inside a sphere inscribed
{{unreferenced, date=August 2012
An inscribed triangle of a circle
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...

in a cube can be approximated as 52.4% of the volume of the cube, since , where is the diameter of the sphere and also the length of a side of the cube and ≈ 0.5236. For example, a sphere with diameter 1m has 52.4% the volume of a cube with edge length 1m, or about 0.524 mSurface area

Thesurface area
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...

of a sphere of radius is:
:$A\; =\; 4\backslash pi\; r^2.$
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

first derived this formula from the fact that the projection to the lateral surface of a circumscribe
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...

d cylinder is area-preserving. Another approach to obtaining the formula comes from the fact that it equals the derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

of the formula for the volume with respect to because the total volume inside a sphere of radius can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius . At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius is simply the product of the surface area at radius and the infinitesimal thickness.
At any given radius , the incremental volume () equals the product of the surface area at radius () and the thickness of a shell ():
:$\backslash delta\; V\; \backslash approx\; A(r)\; \backslash cdot\; \backslash delta\; r.$
The total volume is the summation of all shell volumes:
:$V\; \backslash approx\; \backslash sum\; A(r)\; \backslash cdot\; \backslash delta\; r.$
In the limit as approaches zero this equation becomes:
:$V\; =\; \backslash int\_0^r\; A(r)\; \backslash ,\; dr.$
Substitute :
:$\backslash frac43\backslash pi\; r^3\; =\; \backslash int\_0^r\; A(r)\; \backslash ,\; dr.$
Differentiating both sides of this equation with respect to yields as a function of :
:$4\backslash pi\; r^2\; =\; A(r).$
This is generally abbreviated as:
:$A\; =\; 4\backslash pi\; r^2,$
where is now considered to be the fixed radius of the sphere.
Alternatively, the area element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV = ...

on the sphere is given in spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

by . In Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

, the area element is
: $dS=\backslash frac\backslash prod\_dx\_,\backslash ;\backslash forall\; k.$
The total area can thus be obtained by integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...

:
:$A\; =\; \backslash int\_0^\; \backslash int\_0^\backslash pi\; r^2\; \backslash sin\backslash theta\; \backslash ,\; d\backslash theta\; \backslash ,\; d\backslash varphi\; =\; 4\backslash pi\; r^2.$
The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...

locally minimizes surface area.
The surface area relative to the mass of a ball is called the specific surface area
Specific surface area (SSA) is a property of solids defined as the total surface area of a material per unit of mass, (with units of m2/kg or m2/g) or solid or bulk volume (units of m2/m3 or m−1).
It is a physical value that can be used to det ...

and can be expressed from the above stated equations as
:$\backslash mathrm\; =\; \backslash frac\; =\; \backslash frac,$
where is the density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

(the ratio of mass to volume).
Other geometric properties

A sphere can be constructed as the surface formed by rotating acircle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

about any of its diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...

s; this is essentially the traditional definition of a sphere as given in Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...

. Since a circle is a special type of ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

, a sphere is a special type of ellipsoid of revolution
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has c ...

. Replacing the circle with an ellipse rotated about its major axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...

, the shape becomes a prolate spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...

; rotated about the minor axis, an oblate spheroid.
A sphere is uniquely determined by four points that are not coplanar
In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. Howe ...

. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc. This property is analogous to the property that three non-collinear points determine a unique circle in a plane.
Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.
By examining the common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres. Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point)..
The angle between two spheres at a real point of intersection is the dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection. They intersect at right angles (are orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.
Pencil of spheres

If and are the equations of two distinct spheres then :$s\; f(x,y,z)\; +\; t\; g(x,y,z)\; =\; 0$ is also the equation of a sphere for arbitrary values of the parameters and . The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.''Eleven properties of the sphere''

In their book ''Geometry and the Imagination'',David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...

, which can be thought of as a sphere with infinite radius. These properties are:
# ''The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.''
#: The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar result
A result (also called upshot) is the final consequence of a sequence of actions or events expressed qualitatively or quantitatively. Possible results include advantage, disadvantage, gain, injury, loss, value and victory. There may be a range ...

of Apollonius of Perga
Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος, Apollṓnios ho Pergaîos; la, Apollonius Pergaeus; ) was an Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the contribution ...

for the plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...

.
# ''The contours and plane sections of the sphere are circles.''
#: This property defines the sphere uniquely.
# ''The sphere has constant width and constant girth.''
#: The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the Meissner body
The Reuleaux tetrahedron is the intersection of four balls of radius ''s'' centered at the vertices of a regular tetrahedron with side length ''s''. The spherical surface of the ball centered on each vertex passes through the other three verti ...

. The girth of a surface is the circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to ...

of the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.
# ''All points of a sphere are umbilic
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...

s.''
#: At any point on a surface a normal direction is at right angles to the surface because on the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane that contains the normal with the surface will form a curve that is called a ''normal section,'' and the curvature of this curve is the ''normal curvature''. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by ...

s. Any closed surface will have at least four points called ''umbilical point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are eq ...

s''. At an umbilic all the sectional curvatures are equal; in particular the principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by ...

s are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.
#: For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.
# ''The sphere does not have a surface of centers.''
#: For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along on the normal line. For example, the two centers corresponding to the maximum and minimum sectional curvatures are called the ''focal points'', and the set of all such centers forms the focal surface
For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at th ...

.
#: For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special:
#: * For channel surface
In geometry and topology, a channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its '' directrix''. If the radii of the generating spheres are constant, the canal surface is ca ...

s one sheet forms a curve and the other sheet is a surface
#: * For cones
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines conn ...

, cylinders, tori and cyclides both sheets form curves.
#: * For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere.
# ''All geodesics of the sphere are closed curves.''
#: Geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

are curves on a surface that give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. Many other surfaces share this property.
# ''Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.''
#: It follows from isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...

. These properties define the sphere uniquely and can be seen in soap bubble
A soap bubble is an extremely thin film of soap or detergent and water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact wi ...

s: a soap bubble will enclose a fixed volume, and surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...

minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape). It can also be seen in planets and stars where gravity minimizes surface area for large celestial bodies.
# ''The sphere has the smallest total mean curvature among all convex solids with a given surface area.''
#: The mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...

is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere.
# ''The sphere has constant mean curvature.''
#: The sphere is the only imbedded surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

s have constant mean curvature.
# ''The sphere has constant positive Gaussian curvature.''
#: Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...

is the product of the two principal curvatures. It is an intrinsic property that can be determined by measuring length and angles and is independent of how the surface is embedded in space. Hence, bending a surface will not alter the Gaussian curvature, and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries, and the sphere is the only surface that lacks a boundary with constant, positive Gaussian curvature. The pseudosphere
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius is a surface in \mathbb^3 having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surface ...

is an example of a surface with constant negative Gaussian curvature.
# ''The sphere is transformed into itself by a three-parameter family of rigid motions.''
#: Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189 ...

). Therefore, a three-parameter family of rotations exists such that each rotation transforms the sphere onto itself; this family is the rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...

. The plane is the only other surface with a three-parameter family of transformations (translations along the - and -axes and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the surfaces of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...

and helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known.
Description
It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarit ...

s are the only surfaces with a one-parameter family.
Treatment by area of mathematics

Spherical geometry

The basic elements ofEuclidean plane geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...

are points and lines
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...

. On the sphere, points are defined in the usual sense. The analogue of the "line" is the geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

, which is a great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...

; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...

shows that the shortest path between two points lying on the sphere is the shorter segment of the great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...

that includes the points.
Many theorems from classical geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's postulate
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

s, including the parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segment ...

. In spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...

, angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...

s are defined between great circles. Spherical trigonometry differs from ordinary trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...

in many respects. For example, the sum of the interior angles of a spherical triangle
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...

always exceeds 180 degrees. Also, any two similar spherical triangles are congruent.
Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e. the diameter) are called ''antipodal points''—on the sphere, the distance between them is exactly half the length of the circumference. Any other (i.e. not antipodal) pair of distinct points on a sphere
* lie on a unique great circle,
* segment it into one minor (i.e. shorter) and one major (i.e. longer) arc, and
* have the minor arc's length be the ''shortest distance'' between them on the sphere.
Spherical geometry is a form of elliptic geometry
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...

, which together with hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...

makes up non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

.
Differential geometry

The sphere is asmooth surface
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspectives: ...

with constant Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...

at each point equal to . As per Gauss's Theorema Egregium
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...

, this curvature is independent of the sphere's embedding in 3-dimensional space. Also following from Gauss, a sphere cannot be mapped to a plane while maintaining both areas and angles. Therefore, any map projection
In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitud ...

introduces some form of distortion.
A sphere of radius has area element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV = ...

$dA\; =\; r^2\; \backslash sin\; \backslash theta\backslash ,\; d\backslash theta\backslash ,\; d\backslash varphi$. This can be found from the volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form
:dV = \ ...

in integral surface
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
Name
Integral curves are known by various other names, depending on the nature and interpret ...

of the following differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

:
:$x\; \backslash ,\; dx\; +\; y\; \backslash ,\; dy\; +\; z\; \backslash ,\; dz\; =\; 0.$
This equation reflects that the position vector and tangent plane
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

at a point are always orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

to each other. Furthermore, the outward-facing normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...

is equal to the position vector scaled by .
In Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...

, the filling area conjecture In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introducing shortcuts between its points.
Definitions ...

states that the hemisphere is the optimal (least area) isometric filling of the Riemannian circle
In metric space theory and Riemannian geometry, the Riemannian circle is a great circle with a characteristic length. It is the circle equipped with the ''intrinsic'' Riemannian metric of a compact one-dimensional manifold of total length 2, or ...

.
Topology

Intopology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, an -sphere is defined as a space homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

to the boundary of an -ball; thus, it is homeomorphic to the Euclidean -sphere, but perhaps lacking its metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...

.
* A 0-sphere is a pair of points with the discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...

.
* A 1-sphere is a circle (up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...

homeomorphism); thus, for example, (the image of) any knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

is a 1-sphere.
* A 2-sphere is an ordinary sphere (up to homeomorphism); thus, for example, any spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...

is a 2-sphere.
The -sphere is denoted . It is an example of a compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...

topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...

without boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...

. A sphere need not be smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...

; if it is smooth, it need not be diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...

to the Euclidean sphere (an exotic sphere).
The sphere is the inverse image of a one-point set under the continuous function , so it is closed; is also bounded, so it is compact by the Heine–Borel theorem.
Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...

with possible self-intersections but without creating any creases, in a process called sphere eversion
In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word '' eversion'' means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere in ...

.
The antipodal quotient of the sphere is the surface called the real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...

, which can also be thought of as the Northern Hemisphere
The Northern Hemisphere is the half of Earth that is north of the Equator. For other planets in the Solar System, north is defined as being in the same celestial hemisphere relative to the invariable plane of the solar system as Earth's Nort ...

with antipodal points of the equator identified.
Curves on a sphere

Circles

Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty. Great circles are the intersection of the sphere with a plane passing through the center of a sphere: others are called small circles. More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with asurface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...

whose axis contains the center of the sphere (are ''coaxial'') consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty.
Loxodrome

Innavigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...

, a rhumb line or loxodrome is an arc crossing all meridians of longitude
Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...

at the same angle. Loxodromes are the same as straight lines in the Mercator projection
The Mercator projection () is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because it is unique in representing north as up and sou ...

. A rhumb line is not a spherical spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:
A Clelia curve is a curve on a sphere for which the

circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

of radius ''r''
* : a 2-sphere is an ordinary sphere
* : a 3-sphere is a sphere in 4-dimensional Euclidean space.
Spheres for are sometimes called

ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

, even a large sphere may be an empty set. For example, in with

File:Einstein gyro gravity probe b.jpg, An image of one of the most accurate human-made spheres, as it New Scientist , Technology , Roundest objects in the world created

File:King of spades- spheres.jpg, Deck of playing cards illustrating engineering instruments, England, 1702.

Surface area of sphere proof

{{Authority control Differential geometry Differential topology Elementary geometry Elementary shapes Homogeneous spaces Surfaces Topology

longitude
Longitude (, ) is a geographic coordinate that specifies the east–west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter l ...

$\backslash varphi$ and the colatitude
In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude. Here Southern latitudes are defined to be negative, and as a result the colatitude is a non- ...

$\backslash theta$ satisfy the equation
:$\backslash varphi=c\backslash ;\backslash theta,\; \backslash quad\; c>0$.
Special cases are: Viviani's curve
In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and p ...

($c=1$) and spherical spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:Seiffert's spiral
Seiffert's spherical spiral is a curve on a sphere made by moving on the sphere with constant speed and angular velocity with respect to a fixed diameter. If the selected diameter is the line from the north pole to the south pole, then the requir ...

. Clelia curves approximate the path of satellites in polar orbit
A polar orbit is one in which a satellite passes above or nearly above both poles of the body being orbited (usually a planet such as the Earth, but possibly another body such as the Moon or Sun) on each revolution. It has an inclination of about ...

.
Spherical conics

The analog of aconic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

on the sphere is a spherical conic
In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in th ...

, a quartic curve which can be defined in several equivalent ways, including:
* as the intersection of a sphere with a quadratic cone whose vertex is the sphere center;
* as the intersection of a sphere with an elliptic or hyperbolic cylinder whose axis passes through the sphere center;
* as the locus of points whose sum or difference of great-circle distance
The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle.
It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...

s from a pair of foci
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...

is a constant.
Many theorems relating to planar conic sections also extend to spherical conics.
Intersection of a sphere with a more general surface

If a sphere is intersected by another surface, there may be more complicated spherical curves. ; Example: sphere – cylinder The intersection of the sphere with equation $\backslash ;\; x^2+y^2+z^2=r^2\backslash ;$ and the cylinder with equation $\backslash ;(y-y\_0)^2+z^2=a^2,\; \backslash ;\; y\_0\backslash ne\; 0\backslash ;$ is not just one or two circles. It is the solution of the non-linear system of equations :$x^2+y^2+z^2-r^2=0$ :$(y-y\_0)^2+z^2-a^2=0\backslash \; .$ (seeimplicit curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly ''x'' and ''y''. For example, the unit circle is defined by the implicit equation x^2+y^2=1. In general, every implic ...

and the diagram)
Generalizations

Ellipsoids

Anellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...

is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under an affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...

. An ellipsoid bears the same relationship to the sphere that an ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

does to a circle.
Dimensionality

Spheres can be generalized to spaces of any number ofdimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

s. For any natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...

, an ''-sphere,'' often denoted , is the set of points in ()-dimensional Euclidean space that are at a fixed distance from a central point of that space, where is, as before, a positive real number. In particular:
* : a 0-sphere consists of two discrete points, and
* : a 1-sphere is a hypersphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...

s.
The -sphere of unit radius centered at the origin is denoted and is often referred to as "the" -sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.
Metric spaces

More generally, in ametric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

, the sphere of center and radius is the set of points such that .
If the center is a distinguished point that is considered to be the origin of , as in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...

.
Unlike a Euclidean metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore oc ...

, a sphere of radius is nonempty only if can be written as sum of squares of integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s.
An octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

is a sphere in taxicab geometry
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian co ...

, and a cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...

is a sphere in geometry using the Chebyshev distance
In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension. It is na ...

.
History

The geometry of the sphere was studied by the Greeks. ''Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...

'' defines the sphere in book XI, discusses various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII. Euclid does not include the area and volume of a sphere, only a theorem that the volume of a sphere varies as the third power of its diameter, probably due to Eudoxus of Cnidus
Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...

. The volume and area formulas were first determined in Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

's ''On the Sphere and Cylinder
''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of t ...

'' by the method of exhaustion
The method of exhaustion (; ) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area bet ...

. Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume.
Archimedes wrote about the problem of dividing a sphere into segments whose volumes are in a given ratio, but did not solve it. A solution by means of the parabola and hyperbola was given by Dionysodorus
Dionysodorus of Caunus ( grc-gre, Διονυσόδωρος ὁ Καύνειος, c. 250 BC – c. 190 BC) was an ancient List of Greek mathematicians, Greek mathematician.
Life and work
Little is known about the life of Dionysodorus. Pliny the E ...

. A similar problem — to construct a segment equal in volume to a given segment, and in surface to another segment — was solved later by al-Quhi
(; fa, ابوسهل بیژن کوهی ''Abusahl Bijan-e Koohi'') was a Persian mathematician, physicist and astronomer. He was from Kuh (or Quh), an area in Tabaristan, Amol, and flourished in Baghdad in the 10th century. He is considered one o ...

.
Gallery

refracts
In physics, refraction is the redirection of a wave as it passes from one transmission medium, medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most common ...

the image of Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

in the background. This sphere was a fused quartz
Fused quartz, fused silica or quartz glass is a glass consisting of almost pure silica (silicon dioxide, SiO2) in amorphous (non-crystalline) form. This differs from all other commercial glasses in which other ingredients are added which change ...

gyroscope
A gyroscope (from Ancient Greek γῦρος ''gŷros'', "round" and σκοπέω ''skopéō'', "to look") is a device used for measuring or maintaining orientation and angular velocity. It is a spinning wheel or disc in which the axis of rota ...

for the Gravity Probe B
Gravity Probe B (GP-B) was a satellite-based experiment to test two unverified predictions of general relativity: the geodetic effect and frame-dragging. This was to be accomplished by measuring, very precisely, tiny changes in the direction of ...

experiment, and differs in shape from a perfect sphere by no more than 40 atoms (less than 10nm) of thickness. It was announced on 1 July 2008 that Australia
Australia, officially the Commonwealth of Australia, is a Sovereign state, sovereign country comprising the mainland of the Australia (continent), Australian continent, the island of Tasmania, and numerous List of islands of Australia, sma ...

n scientists had created even more nearly perfect spheres, accurate to 0.3nm, as part of an international hunt to find a new global standard kilogram.King of spades
The king of spades is a playing card in the standard 52-card deck.
King of Spades may also refer to:
* ''King of Spades'' (novel), a 1966 novel by Frederick Manfred, the fourth book in ''The Buckskin Man Tales'' series
* King of Spades, a fiction ...

: Spheres
Regions

* Hemisphere *Spherical cap
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a ...

* Spherical lune
In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an example of a digon, θ, with dihedral angle θ. The word "lune" derives from ''luna'', the ...

* Spherical polygon
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...

* Spherical sector
In geometry, a spherical sector, also known as a spherical cone, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. It can be described as the union of a spherical cap and the cone formed ...

* Spherical segment
In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes.
It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum.
The surface of t ...

* Spherical wedge
In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's ''base''). The angle between the radii lying within the bounding semidisks is the dihedral . If is a semi ...

* Spherical zone
In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes.
It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum.
The surface of the ...

See also

* 3-sphere *Affine sphere In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point. The term affine sphere is used because they play an analogous role in affine differential geometr ...

* Alexander horned sphere
The Alexander horned sphere is a pathological object in topology discovered by .
Construction
The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting ...

* Celestial spheres
The celestial spheres, or celestial orbs, were the fundamental entities of the cosmology, cosmological models developed by Plato, Eudoxus of Cnidus, Eudoxus, Aristotle, Ptolemy, Copernicus, and others. In these celestial models, the diurnal m ...

* Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...

* Directional statistics Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Euclidean space, R''n''), axes (lines through the origin in R''n'') or rotations in R''n''. M ...

* Dyson sphere
A Dyson sphere is a hypothetical megastructure that completely encompasses a star and captures a large percentage of its solar power output. The concept is a thought experiment that attempts to explain how a spacefaring civilization would meet ...

* Gauss map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that '' ...

* Hand with Reflecting Sphere
''Hand with Reflecting Sphere'', also known as ''Self-Portrait in Spherical Mirror'', is a lithograph by Dutch artist M. C. Escher, first printed in January 1935. The piece depicts a hand holding a reflective sphere. In the reflection, most ...

, M.C. Escher
Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made Mathematics and art, mathematically inspired woodcuts, lithography, lithographs, and mezzotints.
Despite wide popular interest, Escher was for ...

self-portrait drawing illustrating reflection and the optical properties of a mirror sphere
* Hoberman sphere
A Hoberman sphere is an isokinetic structure patented by Chuck Hoberman that resembles a geodesic dome, but is capable of folding down to a fraction of its normal size by the scissor-like action of its joints. Colorful plastic versions have beco ...

* Homology sphere
* Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure o ...

* Homotopy sphere
In algebraic topology, a branch of mathematics, a ''homotopy sphere'' is an ''n''-manifold that is homotopy equivalent to the ''n''-sphere. It thus has the same homotopy groups and the same homology groups as the ''n''-sphere, and so every homotop ...

* Lenart Sphere Lenart may refer to:
* Municipality of Lenart, Slovenia
* Lenart v Slovenskih Goricah, the seat of the Municipality of Lenart, Slovenia
* Lenart Regional Gifted Center, United States, school
* Lénárt sphere, an educational model for spherical ...

* Napkin ring problem
In geometry, the napkin-ring problem involves finding the volume of a "band" of specified height around a sphere, i.e. the part that remains after a hole in the shape of a circular cylinder is drilled through the center of the sphere. It is a co ...

* Orb (optics)
In photography, backscatter (also called near-camera reflection) is an optical phenomenon resulting in typically circular artifacts on an image, due to the camera's flash being reflected from unfocused motes of dust, water droplets, or other ...

* Pseudosphere
In geometry, a pseudosphere is a surface with constant negative Gaussian curvature.
A pseudosphere of radius is a surface in \mathbb^3 having curvature in each point. Its name comes from the analogy with the sphere of radius , which is a surface ...

* Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...

* Solid angle
In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.
The poi ...

* Sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing p ...

* Spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...

* Spherical cow
Comic of a spherical cow as illustrated by a 1996 meeting of the American Astronomical Association, in reference to astronomy modeling
The spherical cow is a humorous metaphor for highly simplified scientific models of complex phenomena. Origina ...

* Spherical helix, tangent indicatrix In differential geometry, the tangent indicatrix of a closed space curve is a curve on the unit sphere intimately related to the curvature of the original curve. Let \gamma(t) be a closed curve with nowhere-vanishing tangent vector \dot. Then the ta ...

of a curve of constant precession
* Spherical polyhedron
In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhe ...

* Sphericity
Sphericity is a measure of how closely the shape of an object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed a ...

* Tennis ball theorem
In geometry, the tennis ball theorem states that any smooth curve on the surface of a sphere that divides the sphere into two equal-area subsets without touching or crossing itself must have at least four inflection points, points at which the cur ...

* Zoll sphere
Notes and references

Notes

References

Further reading

* . * * . * . * . *External links

* Mathematica/Uniform Spherical DistributionSurface area of sphere proof

{{Authority control Differential geometry Differential topology Elementary geometry Elementary shapes Homogeneous spaces Surfaces Topology