Solid geometry

Short description: Field of mathematics dealing with three-dimensional Euclidean spaces

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

"Solid object" redirects here. It is not to be confused with Rigid object.

Hyperboloid of one sheet

Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space).[1] A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its interior.

Solid geometry deals with the measurements of volumes of various solids, including pyramids, prisms, cubes (and other polyhedrons), cylinders, cones (including truncated) and other solids of revolution.[2]

## Contents

* 1 History
* 2 Topics
* 3 List of solid figures
* 4 Techniques
* 5 Applications
* 6 See also
* 7 Notes
* 8 References

## 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.

## List of solid figures

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

Whereas a sphere is the surface of a ball, for other solid figures it is sometimes ambiguous whether the term refers to the surface of the figure or the volume enclosed therein, notably for a cylinder.

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

| 90px
Rhombohedron |

* A parallelepiped where all edges are the same length
* A cube, except that its faces are not squares but rhombi

| 90px
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 | 80px
Small stellated dodecahedron | 80px
Toroidal polyhedron
Uniform polyhedron | Regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other)  | 80px 80px
(Regular)
Tetrahedron and Cube | 80px
Uniform
Snub dodecahedron
Pyramid | A polyhedron comprising an n-sided polygonal base and a vertex point  | 90px square pyramid
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  | 90px hexagonal prism
Antiprism | A polyhedron comprising an n-sided polygonal base, a second base translated and rotated.sides]] of the two bases  | 90px square antiprism
Bipyramid | A polyhedron comprising an n-sided polygonal center with two apexes.  | 90px triangular bipyramid
Trapezohedron | A polyhedron with 2n kite faces around an axis, with half offsets  | 80px tetragonal trapezohedron
Cone | Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex | 120px
A right circular cone and an oblique circular cone
Cylinder | Straight parallel sides and a circular or oval cross section  | 80px
A solid elliptic cylinder  | 120px
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 | 150px
Examples of ellipsoids  | x2a2+y2b2+z2c2=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] | 90px
Hyperboloid | A surface that is generated by rotating a hyperbola around one of its principal axes | 80px

## 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

* Euclidean geometry
* Shape
* Solid modeling
* Surface

## 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. https://archive.org/details/polytopessymmetr0000robe.
5. ↑ Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. https://archive.org/details/elementssynthet01dupugoog. Retrieved December 1, 2018.
6. ↑ Weisstein, Eric W.. "Lemon". http://mathworld.wolfram.com/Lemon.html.

## References

* Robert Baldwin Hayward (1890) The Elements of Solid Geometry via Internet Archive
* Kiselev, A. P. (2008). Geometry. Book II. Stereometry. Sumizdat.

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