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Solid geometry or stereometry is the of three-dimensional (3D space). The Britannica Guide to Geometry, Britannica Educational Publishing, 2010, pp. 67–68. A solid figure is the region of 3D space bounded by a ; for example, a solid ball consists of a and its interior.

Solid geometry deals with the of of various solids, including pyramids, prisms (and other ), , cylinders, cones (and )..


History
The dealt with the , but the pyramid, prism, cone and cylinder were not studied until the . 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 .Paraphrased and taken in part from the 1911 Encyclopædia Britannica.


Topics
Basic topics in solid geometry and stereometry include:

Advanced topics include:

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


List of solid figures
Whereas a 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 .
+ Major types of shapes that either constitute or define a volume. ! Figure !! Definitions !! colspan=2Images
  • A with six faces (), each of which is a parallelogram
  • A hexahedron with three pairs of parallel faces
  • A prism of which the base is a
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  • A convex polyhedron bounded by six faces, whose is the same as that of a
    (1984). 9780521277396, Cambridge University Press. .
  • Some sources also require that each of the faces is a (so each pair of adjacent faces meets in a ). This more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular , right rectangular prism, or rectangular .
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Flat faces, straight edges and sharp corners or vertices
Small stellated dodecahedron

Toroidal polyhedron
Uniform polyhedron as faces and is vertex-transitive (i.e., there is an mapping any vertex onto any other)
(Regular)
and

Uniform
Snub dodecahedron
A comprising an n-sided base and a vertex point
A comprising an n-sided base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all ) joining corresponding sides of the two bases
A comprising an n-sided base, a second base translated and rotated.sides]] of the two bases
A comprising an n-sided center with two apexes.triangular bipyramid
A with 2 n kite faces around an axis, with half offsetstetragonal trapezohedron
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
Straight parallel sides and a circular or oval cross section
A solid elliptic cylinder

A right and an oblique circular cylinder
A surface that may be obtained from a by deforming it by means of directional scalings, or more generally, of an affine transformation
Examples of ellipsoids
{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1:
(top, a=b=c=4),
(bottom left, a=b=5, c=3),
tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3)]]
LemonA lens (or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc)
A surface that is generated by rotating a 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


Notes

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