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In classical
geometry Geometry (from the grc, γεωμετρία; ''geo-'' "earth", ''-metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and ...
, a radius of a
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 constant. T ...
or
sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its " ...
is any of the
line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry, a line segment is a part of a line that is bounded by two distinct end poi ...
s from its
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts * Graph center, the set of all vertices of minimum eccentricity * Center (ring theory), related to ring theory Pl ...
to its
perimeter A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length (one-dimensional). The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical ...
, and in more modern usage, it is also their length. The name comes from the
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became the dominant language ...
''radius'', meaning ray but also the spoke of a chariot wheel.Definition of Radius
at dictionary.reference.com. Accessed on 2009-08-08.
The plural of radius can be either ''radii'' (from the Latin plural) or the conventional English plural ''radiuses''. The typical abbreviation and mathematical variable name for radius is r. By extension, the
diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre 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 th ...
d is defined as twice the radius:Definition of radius
at mathwords.com. Accessed on 2009-08-08.
:d \doteq 2r \quad \Rightarrow \quad r = \frac d 2. If an object does not have a center, the term may refer to its circumradius, the radius of its
circumscribed circle 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 polygon ...
or
circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing. As in the case of two-dimensional circumscri ...
. In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The
inradius (I), excircles, excenters (J_A, J_B, J_C), internal angle bisectors and external angle bisectors. The green triangle is the excentral triangle. In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the tri ...
of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity. For
regular polygon In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex or star. In the limit, a sequence of regular polygon ...
s, the radius is the same as its circumradius.Barnett Rich, Christopher Thomas (2008), ''Schaum's Outline of Geometry'', 4th edition, 326 pages. McGraw-Hill Professional. ,
Online version
accessed on 2009-08-08.
The inradius of a regular polygon is also called
apothem Apothem of a hexagon The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to ...
. In
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''vertices'' (also called ''nodes'' or ''points'') which are connec ...
, the radius of a graph is the minimum over all vertices ''u'' of the maximum distance from ''u'' to any other vertex of the graph.Jonathan L. Gross, Jay Yellen (2006), ''Graph theory and its applications''. 2nd edition, 779 pages; CRC Press. , 9781584885054
Online version
accessed on 2009-08-08.
The radius of the circle with
perimeter A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length (one-dimensional). The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical ...
(
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 a l ...
) ''C'' is :r = \frac C .


Formula

For many geometric figures, the radius has a well-defined relationship with other measures of the figure.


Circles

The radius of a circle with
area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of ...

area
is : r = \sqrt. The radius of the circle that passes through the three non-
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned obj ...
points , , and is given by : r=\frac, where is the angle . This formula uses the
law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles. According to the law, : \frac \,=\, \frac \,=\, \frac \,=\, 2R, wher ...
. If the three points are given by their coordinates , , and , the radius can be expressed as : r = \frac .


Regular polygons

The radius of a regular polygon with sides of length is given by , where R_n = 1\left/\left(2 \sin \frac\pi n \right)\right. . Values of for small values of are given in the table. If then these values are also the radii of the corresponding regular polygons.


Hypercubes

The radius of a ''d''-dimensional
hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpen ...

hypercube
with side ''s'' is : r = \frac\sqrt.


Use in coordinate systems


Polar coordinates

The polar coordinate system is a two-
dimension thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to s ...
al
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is signif ...
in which each
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Points, ...
on a plane is determined by a
distance Distance is a numerical measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). The distance from a point A to ...
from a fixed point and an
angle In Euclidean geometry, an angle is the figure formed by two rays, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are als ...
from a fixed direction. The fixed point (analogous to the origin of a Cartesian system) is called the ''pole'', and the ray from the pole in the fixed direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'' or ''radius'', and the angle is the ''angular coordinate'', ''polar angle'', or ''
azimuth An azimuth (; from Arabic اَلسُّمُوت ''as-sumūt'', 'the directions', the plural form of the Arabic noun السَّمْت ''as-samt'', meaning 'the direction') is an angular measurement in a spherical coordinate system. The vector from ...
''.


Cylindrical coordinates

In the cylindrical coordinate system, there is a chosen reference axis and a chosen reference plane perpendicular to that axis. The ''origin'' of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is variously called the ''cylindrical'' or ''longitudinal'' axis, to differentiate it from the ''polar axis'', which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction. The distance from the axis may be called the ''radial distance'' or ''radius'', while the angular coordinate is sometimes referred to as the ''angular position'' or as the ''azimuth''. The radius and the azimuth are together called the ''polar coordinates'', as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane. The third coordinate may be called the ''height'' or ''altitude'' (if the reference plane is considered horizontal), ''longitudinal position'', or ''axial position''. " ../nowiki>where ''r'', ''θ'', and ''z'' are cylindrical coordinates ../nowiki> as a function of axial position ../nowiki>"


Spherical coordinates

In a spherical coordinate system, the radius describes the distance of a point from a fixed origin. Its position if further defined by the polar angle measured between the radial direction and a fixed zenith direction, and the azimuth angle, the angle between the orthogonal projection of the radial direction on a reference plane that passes through the origin and is orthogonal to the zenith, and a fixed reference direction in that plane.


See also

*
Bend radiusBend radius, which is measured to the inside curvature, is the minimum radius one can bend a pipe, tube, sheet, cable or hose without kinking it, damaging it, or shortening its life. The ''smaller'' the bend radius, the ''greater'' is the material fl ...
*
Filling radiusIn Riemannian geometry, the filling radius of a Riemannian manifold ''X'' is a metric invariant of ''X''. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizi ...
in Riemannian geometry *
Radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series converges absolutely and uniforml ...
* Radius of convexity *
Radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of ...

Radius of curvature
*
Radius of gyrationRadius of gyration or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated ...
*
Semidiameter In geometry, the semidiameter or semi-diameter of a set of points may be one half of its diameter; or, sometimes, one half of its extent along a particular direction. Special cases The semi-diameter of a sphere, circle, or interval is the same thi ...


References

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