Solid geometry
In mathematics, solid geometry is the traditional name for the geometry of threedimensional Euclidean space[1] (i.e., 3D geometry).
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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.[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 onethird 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
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 


Rhombohedron 


Cuboid 


Polyhedron  Flat polygonal faces, straight edges and sharp corners or vertices  
Uniform polyhedron  Regular polygons as faces and is vertextransitive (i.e., there is an isometry mapping any vertex onto any other) 
 
Prism  A polyhedron comprising an nsided 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  
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  
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
 Surface
 Surface area
 Archimedes
Notes
 The Britannica Guide to Geometry, Britannica Educational Publishing, 2010, pp. 67–68.
 Kiselev 2008.
 Paraphrased and taken in part from the 1911 Encyclopædia Britannica.
 Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. ISBN 9780521277396.
 Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018.
 Weisstein, Eric W. "Lemon". Wolfram MathWorld. Retrieved 20191104.
References
 Kiselev, A. P. (2008). Geometry. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.