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<h2 id="history">History</h2> <p>The <a href="/facts/Pythagoreanism/25BjMN2z">Pythagoreans</a> dealt with the <a href="/facts/Regular_solid/jraREXFD">regular solids</a>, but the pyramid, prism, cone and cylinder were not studied until the <a href="/facts/Platonism/0mvr2x2p">Platonists</a>. <a href="/facts/Eudoxus_of_Cnidus/FgPAy5L9">Eudoxus</a> 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 <a href="/facts/Radius/9Gr9deYD">radius</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="topics">Topics</h2> <p>Basic topics in solid geometry and stereometry include: </p> <ul><li><a href="/facts/Incidence_(geometry)/dlzk9bUl">incidence</a> of <a href="/facts/Plane_(geometry)/tAUtRFcn">planes</a> and <a href="/facts/Line_(mathematics)/gJqsr4Gj">lines</a></li> <li><a href="/facts/Dihedral_angle/C9WEJsuS">dihedral angle</a> and <a href="/facts/Solid_angle/se8IXufg">solid angle</a></li> <li>the <a href="/facts/Cube_(geometry)/65lUaAqN">cube</a>, <a href="/facts/Cuboid/eLLKX4rh">cuboid</a>, <a href="/facts/Parallelepiped/Qn7cDgSr">parallelepiped</a></li> <li>the <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a> and other <a href="/facts/Pyramid_(geometry)/fCXRgFwu">pyramids</a></li> <li><a href="/facts/Prism_(geometry)/baIRRkCy">prisms</a></li> <li><a href="/facts/Octahedron/G38q92xW">octahedron</a>, <a href="/facts/Dodecahedron/7r519eCl">dodecahedron</a>, <a href="/facts/Icosahedron/wWbpnDYp">icosahedron</a></li> <li><a href="/facts/Cone_(geometry)/2c9GIzYD">cones</a> and <a href="/facts/Cylinder_(geometry)/0wQPPr3k">cylinders</a></li> <li>the <a href="/facts/Sphere/jywYLNM2">sphere</a></li> <li>other <a href="/facts/Quadric/L3cTApd4">quadrics</a>: <a href="/facts/Spheroid/GJyyLxQ7">spheroid</a>, <a href="/facts/Ellipsoid/pe9vm0Sh">ellipsoid</a>, <a href="/facts/Paraboloid/AUdRV2QH">paraboloid</a> and <a href="/facts/Hyperboloid/K5mDV2Nw">hyperboloids</a>.</li></ul> <p>Advanced topics include: </p> <ul><li><a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a> of three dimensions (leading to a proof of <a href="/facts/Desargues%2527_theorem/vJzx0uPh">Desargues' theorem</a> by using an extra dimension)</li> <li>further <a href="/facts/Polyhedra/UC0p9FTF">polyhedra</a></li> <li><a href="/facts/Descriptive_geometry/YbHHQeTI">descriptive geometry</a>.</li></ul> <h2 id="list-of-solid-figures">List of solid figures</h2> <p class="note">For a more complete list and organization, see <a href="/facts/List_of_mathematical_shapes/cpXPlmKY">List of mathematical shapes</a>.</p> <p>Whereas a <a href="/facts/Sphere/jywYLNM2">sphere</a> is the surface of a <a href="/facts/Ball_(mathematics)/f9vpMVMp">ball</a>, 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 <a href="/facts/Cylinder/0wQPPr3k">cylinder</a>. </p> Major types of shapes that either constitute or define a volume.<table><tbody><tr><th>Figure</th><th>Definitions</th><th colspan="2">Images</th></tr><tr><td><a href="/facts/Parallelepiped/Qn7cDgSr">Parallelepiped</a></td><td><ul><li>A <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a> with six faces (<a href="/facts/Hexahedron/UzruYxzd">hexahedron</a>), each of which is a parallelogram</li><li>A hexahedron with three pairs of parallel faces</li><li>A <a href="/facts/Prism_(geometry)/baIRRkCy">prism</a> of which the base is a <a href="/facts/Parallelogram/DRuktq60">parallelogram</a></li></ul></td><td colspan="2"></td></tr><tr><td><a href="/facts/Rhombohedron/S7XSr6Yu">Rhombohedron</a></td><td><ul><li>A <a href="/facts/Parallelepiped/Qn7cDgSr">parallelepiped</a> where all edges are the same length</li><li>A <a href="/facts/Cube/65lUaAqN">cube</a>, except that its faces are not squares but <a href="/facts/Rhombus/T4pU50GF">rhombi</a></li></ul></td><td colspan="2"></td></tr><tr><td><a href="/facts/Cuboid/eLLKX4rh">Cuboid</a></td><td><ul><li>A <a href="/facts/Convex_polyhedron/2pRHcRgC">convex polyhedron</a> bounded by six <a href="/facts/Quadrilateral/QCGqrkp2">quadrilateral</a> faces, whose <a href="/facts/Polyhedral_graph/IKPCDH6r">polyhedral graph</a> is the same as that of a <a href="/facts/Cube/65lUaAqN">cube</a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li><li>Some sources also require that each of the faces is a <a href="/facts/Rectangle/mdLfIGKA">rectangle</a> (so each pair of adjacent faces meets in a <a href="/facts/Right_angle/psaslRq7">right angle</a>). This more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular <a href="/facts/Hexahedron/UzruYxzd">hexahedron</a>, right rectangular prism, or rectangular <a href="/facts/Parallelepiped/Qn7cDgSr">parallelepiped</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul></td><td colspan="2"></td></tr><tr><td><a href="/facts/Polyhedron/UC0p9FTF">Polyhedron</a></td><td>Flat <a href="/facts/Polygon/yAq7RjPz">polygonal</a> <a href="/facts/Face_(geometry)/T2E6BTU4">faces</a>, straight <a href="/facts/Edge_(geometry)/3lFN6sEz">edges</a> and sharp corners or <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a></td><td><a href="/facts/Small_stellated_dodecahedron/lX1zlo6C">Small stellated dodecahedron</a></td><td><a href="/facts/Toroidal_polyhedron/g6SS84k2">Toroidal polyhedron</a></td></tr><tr><td><a href="/facts/Uniform_polyhedron/PHnprqRC">Uniform polyhedron</a></td><td><a href="/facts/Regular_polygon/ws89Nhpt">Regular polygons</a> as <a href="/facts/Face_(geometry)/T2E6BTU4">faces</a> and is <a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a> (i.e., there is an <a href="/facts/Isometry/K5wX8ah6">isometry</a> mapping any vertex onto any other)</td><td> (Regular)<a href="/facts/Tetrahedron/7mkSrl6j">Tetrahedron</a> and <a href="/facts/Cube/65lUaAqN">Cube</a></td><td>Uniform<a href="/facts/Snub_dodecahedron/oouEPVxD">Snub dodecahedron</a></td></tr><tr><td><a href="/facts/Pyramid_(geometry)/fCXRgFwu">Pyramid</a></td><td>A <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a> comprising an <i>n</i>-sided <a href="/facts/Polygon/yAq7RjPz">polygonal</a> <a href="/facts/Base_(geometry)/YkaFWKCF">base</a> and a vertex point</td><td colspan="2"> <a href="/facts/Square_pyramid/wDetDohM">square pyramid</a></td></tr><tr><td><a href="/facts/Prism_(geometry)/baIRRkCy">Prism</a></td><td>A <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a> comprising an <i>n</i>-sided <a href="/facts/Polygon/yAq7RjPz">polygonal</a> <a href="/facts/Base_(geometry)/YkaFWKCF">base</a>, a second base which is a <a href="/facts/Translation_(geometry)/KlpWmnfX">translated</a> copy (rigidly moved without rotation) of the first, and <i>n</i> other <a href="/facts/Face_(geometry)/T2E6BTU4">faces</a> (necessarily all <a href="/facts/Parallelogram/DRuktq60">parallelograms</a>) joining <a href="/facts/Corresponding_sides/UkX6JvjL">corresponding sides</a> of the two bases</td><td colspan="2"> <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prism</a></td></tr><tr><td><a href="/facts/Antiprism/FgP5jIVd">Antiprism</a></td><td>A <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a> comprising an <i>n</i>-sided <a href="/facts/Polygon/yAq7RjPz">polygonal</a> <a href="/facts/Base_(geometry)/YkaFWKCF">base</a>, a second base translated and rotated.sides]] of the two bases</td><td colspan="2"> <a href="/facts/Square_antiprism/4514Q02s">square antiprism</a></td></tr><tr><td><a href="/facts/Bipyramid/MkzrCA0W">Bipyramid</a></td><td>A <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a> comprising an <i>n</i>-sided <a href="/facts/Polygon/yAq7RjPz">polygonal</a> center with two apexes.</td><td colspan="2"> <a href="/facts/Triangular_bipyramid/fU4kIBTV">triangular bipyramid</a></td></tr><tr><td><a href="/facts/Trapezohedron/8QpjlnXe">Trapezohedron</a></td><td>A <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a> with 2<i>n</i> kite faces around an axis, with half offsets</td><td colspan="2"> <a href="/facts/Tetragonal_trapezohedron/Lnv7NUfC">tetragonal trapezohedron</a></td></tr><tr><td><a href="/facts/Cone/2c9GIzYD">Cone</a></td><td>Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the <a href="/facts/Apex_(geometry)/SrMxmHJk">apex</a> or <a href="/facts/Vertex_(geometry)/ID3cea9z">vertex</a></td><td colspan="2">A right circular cone and an oblique circular cone</td></tr><tr><td><a href="/facts/Cylinder/0wQPPr3k">Cylinder</a></td><td>Straight parallel sides and a circular or oval cross section</td><td>A solid elliptic cylinder</td><td>A right and an oblique circular cylinder</td></tr><tr><td><a href="/facts/Ellipsoid/pe9vm0Sh">Ellipsoid</a></td><td>A surface that may be obtained from a <a href="/facts/Sphere/jywYLNM2">sphere</a> by deforming it by means of directional <a href="/facts/Scaling_(geometry)/pFm6fvpu">scalings</a>, or more generally, of an <a href="/facts/Affine_transformation/BD7yDr10">affine transformation</a></td><td>Examples of ellipsoids</td><td> x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 : {\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1:} <i><a href="/facts/Sphere/jywYLNM2">sphere</a></i> (top, a=b=c=4),<p><i><a href="/facts/Spheroid/GJyyLxQ7">spheroid</a></i> (bottom left, a=b=5, c=3),<i>tri-axial</i> ellipsoid (bottom right, a=4.5, b=6, c=3)]]</p></td></tr><tr><td><a href="/facts/Lemon_(geometry)/0HoL5Ixu">Lemon</a></td><td>A <a href="/facts/Lens_(geometry)/mpgyWrWz">lens</a> (or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc)<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></td><td colspan="2"></td></tr><tr><td><a href="/facts/Hyperboloid/K5mDV2Nw">Hyperboloid</a></td><td>A <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a> that is generated by rotating a <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> around one of its <a href="/facts/Hyperbola/6R6ER0mY">principal axes</a></td><td colspan="2"></td></tr></tbody></table> <h2 id="techniques">Techniques</h2> <p>Various techniques and tools are used in solid geometry. Among them, <a href="/facts/Analytic_geometry/3qWP0kM0">analytic geometry</a> and <a href="/facts/Vector_(geometric)/bCLrdksy">vector</a> techniques have a major impact by allowing the systematic use of <a href="/facts/System_of_linear_equations/fk9CFdgp">linear equations</a> and <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> algebra, which are important for higher dimensions. </p> <h2 id="applications">Applications</h2> <p>A major application of solid geometry and stereometry is in <a href="/facts/3D_computer_graphics/zlWW6eaF">3D computer graphics</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a></li> <li><a href="/facts/Shape/nww0x85d">Shape</a></li> <li><a href="/facts/Solid_modeling/12J69aud">Solid modeling</a></li> <li><a href="/facts/Surface_(mathematics)/whF08jvy">Surface</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Robert_Baldwin_Hayward/M6ffBv75">Robert Baldwin Hayward</a> (1890) <a href="https://archive.org/details/elementssolidge00haywgoog/page/n10/mode/2up"><i>The Elements of Solid Geometry</i></a> via <a href="/facts/Internet_Archive/PHStU8Lo">Internet Archive</a></li> <li>Kiselev, A. P. (2008). <i>Geometry</i>. Vol. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>The Britannica Guide to Geometry, Britannica Educational Publishing, 2010, pp. 67–68. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Kiselev 2008. - Kiselev, A. P. (2008). Geometry. Vol. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Paraphrased and taken in part from the 1911 Encyclopædia Britannica. <a href="/wiki/1911_Encyclop%C3%A6dia_Britannica" target="_blank">/wiki/1911_Encyclop%C3%A6dia_Britannica</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Robertson, Stewart Alexander (1984). Polytopes and Symmetry. Cambridge University Press. p. 75. ISBN 9780521277396. <a href="9780521277396" target="_blank">9780521277396</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Dupuis, Nathan Fellowes (1893). Elements of Synthetic Solid Geometry. Macmillan. p. 53. Retrieved December 1, 2018. <a href="https://archive.org/details/elementssynthet01dupugoog" target="_blank">https://archive.org/details/elementssynthet01dupugoog</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Weisstein, Eric W. "Lemon". Wolfram MathWorld. Retrieved 2019-11-04. <a href="http://mathworld.wolfram.com/Lemon.html" target="_blank">http://mathworld.wolfram.com/Lemon.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>