Solid geometry

Not to be confused with the film of the same name.

In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces^[1] (i.e., 3D geometry).

Hyperboloid of one sheet

Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), including pyramids, prisms and other polyhedrons; cubes; cylinders; cones; truncated cones; and balls bounded by spheres.^[2]

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.^[3]

Topics

Basic topics in solid geometry and stereometry include:

• incidence of planes and lines
• dihedral angle and solid angle
• the cube, cuboid, parallelepiped
• the tetrahedron and other pyramids
• prisms
• octahedron, dodecahedron, icosahedron
• cones and cylinders
• the sphere
• other quadrics: spheroid, ellipsoid, paraboloid and hyperboloids.

Advanced topics include:

• projective geometry of three dimensions (leading to a proof of Desargues' theorem by using an extra dimension)
• further polyhedra
• descriptive geometry.

Solid figures

For a more complete list and organization, see List of mathematical shapes.

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.

Figure	Definitions	Images
Parallelepiped	A polyhedron with six faces (hexahedron), each of which is a parallelogram A hexahedron with three pairs of parallel faces A prism of which the base is a parallelogram
Rhombohedron	A parallelepiped where all edges are the same length A cube, except that its faces are not squares but rhombi
Cuboid	A convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube^[4] Some sources also require that each of the faces is a rectangle (so each pair of adjacent faces meets in a right angle). This more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped.^[5]
Polyhedron	Flat polygonal faces, straight edges and sharp corners or vertices	Small stellated dodecahedron   Toroidal polyhedron
Uniform polyhedron	Regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other)	Tetrahedron   Snub dodecahedron
Prism	A 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	A solid elliptic cylinder   A right and an oblique circular cylinder
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 ${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1:}$  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)^[6]
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.

See also

• Ball regions
• Euclidean geometry
• Dimension
• Point
• Planimetry
• Shape
• Lists of shapes
• Solid modeling
• Surface
• Surface area
• Archimedes

Notes

1. The Britannica Guide to Geometry, Britannica Educational Publishing, 2010, pp. 67–68.
2. Kiselev 2008.
3. Paraphrased and taken in part from the 1911 Encyclopædia Britannica.
4. Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. ISBN 9780521277396.
5. Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.
6. Weisstein, Eric W. "Lemon". Wolfram MathWorld. Retrieved 2019-11-04.

References

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