In

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, solid geometry or stereometry is the traditional name for the geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

of three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:παρά#Ancient Greek, παρά, ''par ...

, Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

s (i.e., 3D geometry).
Stereometry deals with the measurement
Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependen ...

s of volume
Volume is a scalar quantity expressing the amount
Quantity or amount is a property that can exist as a multitude
Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

s of various solid figures (or 3D figures), including pyramids
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilater ...

, prisms
An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical shape o ...

and other ; cylinders
A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physi ...

; cones
A cone is a three-dimensional space, three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the Apex (geometry), apex or vertex (geometry), vertex.
A cone is fo ...

; ; and balls
A ball
A ball is a round object (usually spherical, but can sometimes be ovoid
An oval (from Latin ''ovum'', "egg") is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas ( p ...

bounded by sphere
A sphere (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

s.
History

ThePythagoreans
Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spr ...

dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonist
Platonism is the philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of ...

s. Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height. He was probably also the discoverer of a proof that the volume enclosed by a sphere is proportional to the cube of its radius
In classical geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

.Paraphrased and taken in part from the '' 1911 Encyclopædia Britannica''.
Topics

Basic topics in solid geometry and stereometry include: * incidence ofplane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early flying machines include all forms of aircraft studied ...

s and line
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', a 2009 independent film by Nancy Schwartzman
Lite ...

s
* dihedral angle
A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, ele ...

and 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 ...

* the cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

, cuboid
In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid, other sources use "cuboid" to refer to a ...

, parallelepiped
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of f ...

* the tetrahedron
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

and other pyramid
A pyramid (from el, πυραμίς ') is a structure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act ...

s
* prism
A prism
An optical prism is a transparent optics, optical element with flat, polished surfaces that refraction, refract light. At least one surface must be angled—elements with two parallel surfaces are not prisms. The traditional geometrical ...

s
* octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral tri ...

, dodecahedron
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

, icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non-similarity (geometry), similar shapes of icosahedra, some of them ...

* cone
A cone is a three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter
A parameter (from the Ancient Greek language, Ancient Greek wikt:πα ...

s and cylinders
A cylinder (from Greek language, Greek κύλινδρος – ''kulindros'', "roller", "tumbler") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physi ...

* the sphere
A sphere (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

* other quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimension
thumb
, 236px
, The first four spatial dimensions, represented in a two-dimensional picture.
In physics
Physics is the natural science that s ...

s: spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipse
In math ...

, ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a Surface (mathemat ...

, paraboloid
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

and hyperboloid
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

s.
Advanced topics include:
* projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...

of three dimensions (leading to a proof of Desargues' theorem
for the truth of the second.
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective (geometry), perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote ...

by using an extra dimension)
* further polyhedra
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

* descriptive geometry.
Solid figures

Whereas asphere
A sphere (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

is the surface of a ball
A ball is a round object (usually spherical
of a sphere
A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...

, it is sometimes ambiguous whether the term refers to the surface of the figure or the volume enclosed therein, notably for a cylinder
A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base.
This traditi ...

. The following table includes major types of shapes that either constitute or define a volume.
Techniques

Various techniques and tools are used in solid geometry. Among them,analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...

and vector
Vector may refer to:
Biology
*Vector (epidemiology)
In epidemiology
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...

techniques have a major impact by allowing the systematic use of linear equations
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

and matrix
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols, or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the material in between a eukaryoti ...

algebra, which are important for higher dimensions.
Applications

A major application of solid geometry and stereometry is in3D computer graphics
3D computer graphics, sometimes called CGI, 3DCG or three-dimensional computer graphics (in contrast to 2D computer graphics
2D computer graphics is the Computer-generated imagery, computer-based generation of digital images—mostly from t ...

.
See also

* Ball regions *Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandria
Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic
Coptic may refer to:
Afro-Asia
* Copts, an ethnoreligious group mainly in the area of modern ...

* Dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

* Point
Point or points may refer to:
Places
* Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point
Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...

* Planimetry
* Shape
A shape or figure is the form of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type.
A plane shape, two-dimensional s ...

* Lists of shapes
* Surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile.
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...

* Surface area
The surface area of a solid
Solid is one of the four fundamental states of matter
4 (four) is a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of ...

* Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Notes

References

* {{DEFAULTSORT:Solid Geometry *Solid geometry
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...