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Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space). 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 (and other polyhedrons), cubes, cylinders, cones (and truncated cones).
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:
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
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.
| Figure | Definitions | Images | |
|---|---|---|---|
| Parallelepiped |
|  | |
| Rhombohedron |
|  | |
| Cuboid |
|  | |
| 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) |   (Regular) Tetrahedron and Cube |  Unform Snub dodecahedron |
| Pyramid | A polyhedron comprising an n-sided polygonal base and a vertex point |  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 |  hexagonal prism | |
| Antiprism | A polyhedron comprising an n-sided polygonal base, a second base translated and rotated.sides]] of the two bases |  square antiprism | |
| Bipyramid | A polyhedron comprising an n-sided polygonal center with two apexes. |  triangular bipyramid | |
| Trapezohedron | A polyhedron with 2n kite faces around an axis, with half offsets |  tetragonal trapezohedron | |
| 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 |  sphere (top, a=b=c=4), spheroid (bottom left, a=b=5, 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.
See also
- Euclidean geometry
- Shape
- Solid modeling
- Surface
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 2019-11-04.
References
- Robert Baldwin Hayward (1890) The Elements of Solid Geometry via Internet Archive
- Kiselev, A. P. (2008). Geometry. Vol. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.
This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Solid geometry news newspapers books scholar JSTOR May 2014 Learn how and when to remove this message Solid geometry or stereometry is the geometry of three dimensional Euclidean space 3D space 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 Hyperboloid of one sheet Solid geometry deals with the measurements of volumes of various solids including pyramids prisms and other polyhedrons cubes cylinders cones and truncated cones HistoryThe 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 TopicsBasic 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 Plane vs Solid Geometry 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 figuresWhereas 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 ImagesParallelepiped 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 parallelogramRhombohedron A parallelepiped where all edges are the same length A cube except that its faces are not squares but rhombiCuboid A convex polyhedron bounded by six quadrilateral faces whose polyhedral graph is the same as that of a cube 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 Polyhedron Flat polygonal faces straight edges and sharp corners or vertices Small stellated dodecahedron Toroidal polyhedronUniform polyhedron Regular polygons as faces and is vertex transitive i e there is an isometry mapping any vertex onto any other Regular Tetrahedron and Cube Unform Snub dodecahedronPyramid A polyhedron comprising an n sided polygonal base and a vertex point square pyramidPrism 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 hexagonal prismAntiprism A polyhedron comprising an n sided polygonal base a second base translated and rotated sides of the two bases square antiprismBipyramid A polyhedron comprising an n sided polygonal center with two apexes triangular bipyramidTrapezohedron A polyhedron with 2n kite faces around an axis with half offsets tetragonal trapezohedronCone 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 coneCylinder Straight parallel sides and a circular or oval cross section A solid elliptic cylinder A right and an oblique circular cylinderEllipsoid 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 x2a2 y2b2 z2c2 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 axesTechniquesVarious 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 ApplicationsA major application of solid geometry and stereometry is in 3D computer graphics See alsoEuclidean geometry Shape Solid modeling SurfaceNotesThe Britannica Guide to Geometry Britannica Educational Publishing 2010 pp 67 68 Kiselev 2008 Paraphrased and taken in part from the 1911 Encyclopaedia 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 2019 11 04 ReferencesRobert Baldwin Hayward 1890 The Elements of Solid Geometry via Internet Archive Kiselev A P 2008 Geometry Vol Book II Stereometry Translated by Givental Alexander Sumizdat
