# Solid geometry

In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space (i.e., 3D geometry).

Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), including pyramids, prisms and other polyhedrons; cylinders; cones; truncated cones; and balls bounded by spheres.

## History

The Pythagoreans dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. 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.

## Topics

Basic topics in solid geometry and stereometry include:

## Solid figures

Whereas a sphere is the surface of a ball, it is sometimes ambiguous whether the term refers to the surface of the figure or the volume enclosed therein, notably for a cylinder. The following table includes major types of shapes that either constitute or define a volume.

FigureDefinitionsImages
Parallelepiped Rhombohedron Cuboid Polyhedron Flat polygonal faces, straight edges and sharp corners or vertices
Uniform polyhedron Regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other)
PrismA polyhedron comprising an n-sided polygonal base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases Cone Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex A right circular cone and an oblique circular cone
Cylinder Straight parallel sides and a circular or oval cross section
Ellipsoid A surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation Examples of ellipsoids with equation x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 : {\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1:} sphere (top, a=b=c=4), spheroid (bottom left, a=b=5, c=3), tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3)
Lemon A lens (or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc) Hyperboloid A surface that is generated by rotating a hyperbola around one of its principal axes ## Techniques

Various techniques and tools are used in solid geometry. Among them, analytic geometry and vector techniques have a major impact by allowing the systematic use of linear equations and matrix algebra, which are important for higher dimensions.

## Applications

A major application of solid geometry and stereometry is in 3D computer graphics.

## Notes

1. The Britannica Guide to Geometry, Britannica Educational Publishing, 2010, pp. 67–68.
2. Paraphrased and taken in part from the 1911 Encyclopædia Britannica.
3. Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. ISBN 9780521277396.
4. Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.
5. Weisstein, Eric W. "Lemon". Wolfram MathWorld. Retrieved 2019-11-04.

## References

• Kiselev, A. P. (2008). Geometry. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.