Editing Solid geometry

{{Short description|Field of mathematics dealing with three-dimensional Euclidean spaces}}
{{Distinguish|Solid Geometry (film){{!}}the film of the same name}}
{{redirect-distinguish|Solid object|Rigid object}}
{{redirect|Solid surface|the material|Solid surface material}}
{{more citations needed|date=May 2014}}
[[File:Hyperboloid1.png|thumb|237x237px|[[Hyperboloid]] of one sheet]]

'''Solid geometry''' or '''stereometry''' is the [[geometry]] of [[Three-dimensional space|three-dimensional]] [[Euclidean space]] (3D space).<ref>''The Britannica Guide to Geometry'', Britannica Educational Publishing, 2010, pp. 67–68.</ref>
A '''solid figure''' is the [[region (mathematics)|region]] of 3D space bounded by a [[two-dimensional]] [[closed surface]]; for example, a solid [[ball (mathematics)|ball]] consists of a [[sphere]] and its [[Interior (topology)|interior]].

Solid geometry deals with the [[measurement]]s of [[volume]]s of various solids, including [[Pyramid (geometry)|pyramids]], [[Prism (geometry)|prisms]] (and other [[polyhedrons]]), [[cubes]], [[Cylinder (geometry)|cylinders]], [[cone (geometry)|cones]] (and [[Frustum|truncated cones]]).<ref>{{harvnb|Kiselev|2008}}.</ref>

==History==

The [[Pythagoreanism|Pythagoreans]] dealt with the [[regular solid]]s, but the pyramid, prism, cone and cylinder were not studied until the [[Platonism|Platonist]]s. [[Eudoxus of Cnidus|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]].<ref>Paraphrased and taken in part from the ''[[1911 Encyclopædia Britannica]]''.</ref>

==Topics==

Basic topics in solid geometry and stereometry include:
{{Div col|colwidth=27em}}
* [[incidence (geometry)|incidence]] of [[plane (geometry)|plane]]s and [[line (mathematics)|line]]s
* [[dihedral angle]] and [[solid angle]]
* the [[Cube (geometry)|cube]], [[cuboid]], [[parallelepiped]]
* the [[tetrahedron]] and other [[pyramid (geometry)|pyramid]]s
* [[Prism (geometry)|prism]]s
* [[octahedron]], [[dodecahedron]], [[icosahedron]]
* [[cone (geometry)|cone]]s and [[cylinder (geometry)|cylinders]]
* the [[sphere]]
* other [[quadric]]s: [[spheroid]], [[ellipsoid]], [[paraboloid]] and [[hyperboloid]]s.
{{Div col end}}

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|List of mathematical shapes}}
Whereas a [[sphere]] is the surface of a [[ball (mathematics)|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]]. 
{|class="wikitable"
|+ Major types of shapes that either constitute or define a volume.
! Figure !! Definitions !! colspan=2|Images
|-
| [[Parallelepiped]] || 
*A [[polyhedron]] with six faces ([[hexahedron]]), each of which is a parallelogram
*A hexahedron with three pairs of parallel faces
*A [[prism (geometry)|prism]] of which the base is a [[parallelogram]]
|colspan=2|[[Image:Parallelepiped 2013-11-29.svg|90px]]
|-
| [[Rhombohedron]] ||
*A [[parallelepiped]] where all edges are the same length
*A [[cube]], except that its faces are not squares but [[rhombus|rhombi]]
|colspan=2|[[Image:Rhombohedron.svg|90px]]
|-
| [[Cuboid]] ||
*A [[convex polyhedron]] bounded by six [[quadrilateral]] faces, whose [[polyhedral graph]] is the same as that of a [[cube]]<ref>{{cite book |title=Polytopes and Symmetry |url=https://archive.org/details/polytopessymmetr0000robe |url-access=registration |first=Stewart Alexander |last=Robertson |publisher=Cambridge University Press |year=1984 |isbn=9780521277396 |page=[https://archive.org/details/polytopessymmetr0000robe/page/75 75]}}</ref>
*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]]'''.<ref>{{cite book |url=https://archive.org/details/elementssynthet01dupugoog |title=Elements of Synthetic Solid Geometry |first=Nathan Fellowes |last=Dupuis |publisher=Macmillan |year=1893 |page=[https://archive.org/details/elementssynthet01dupugoog/page/n69 53] |access-date=December 1, 2018}}</ref>
|colspan=2|[[File:Cuboid_no_label.svg|90px|Rectangular cuboid]]
|-
| [[Polyhedron]]
| Flat [[polygon]]al [[Face (geometry)|faces]], straight [[Edge (geometry)|edges]] and sharp corners or [[Vertex (geometry)|vertices]]
|[[File:Small stellated dodecahedron.png|80px]]<BR>[[Small stellated dodecahedron]]
|[[File:Hexagonal torus.svg|80px]]<BR>[[Toroidal polyhedron]]
|-
| [[Uniform polyhedron]]
| [[Regular polygon]]s as [[Face (geometry)|faces]] and is [[vertex-transitive]] (i.e., there is an [[isometry]] mapping any vertex onto any other)
|[[File:Uniform polyhedron-33-t0.png|80px]] [[File:Uniform polyhedron-43-t0.png|80px]]<BR>(Regular)<BR>[[Tetrahedron]] and [[Cube]]
|[[File:Uniform polyhedron-53-s012.png|80px]]<BR>Uniform<BR>[[Snub dodecahedron]]
|-
| [[Pyramid (geometry)|Pyramid]] || A [[polyhedron]] comprising an ''n''-sided [[polygon]]al [[base (geometry)|base]] and a vertex point
|colspan=2|[[File:Square pyramid.png|90px]] [[square pyramid]]
|-
| [[Prism (geometry)|Prism]] || A [[polyhedron]] comprising an ''n''-sided [[polygon]]al [[base (geometry)|base]], a second base which is a [[Translation (geometry)|translated]] copy (rigidly moved without rotation) of the first, and ''n'' other [[face (geometry)|faces]] (necessarily all [[parallelogram]]s) joining [[corresponding sides]] of the two bases
|colspan=2|[[Image:Hexagonal Prism BC.svg|90px]] [[hexagonal prism]]
|-
| [[Antiprism]] || A [[polyhedron]] comprising an ''n''-sided [[polygon]]al [[base (geometry)|base]], a second base translated and rotated.sides]] of the two bases
|colspan=2|[[Image:Square antiprism.png|90px]] [[square antiprism]]
|-
| [[Bipyramid]] || A [[polyhedron]] comprising an ''n''-sided [[polygon]]al center with two apexes.
|colspan=2|[[File:Triangular bipyramid.png|90px]] [[triangular bipyramid]]
|-
| [[Trapezohedron]] || A [[polyhedron]] with 2''n'' kite faces around an axis, with half offsets
|colspan=2|[[File:Tetragonal trapezohedron.png|80px]] [[tetragonal trapezohedron]]
|-
| [[Cone]]
| Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the [[Apex (geometry)|apex]] or [[vertex (geometry)|vertex]]
|colspan=2|[[File:Cone 3d.png|120px]]<BR>A right circular cone and an oblique circular cone
|-
| [[Cylinder]]
| Straight parallel sides and a circular or oval cross section
|[[File:Elliptic cylinder abh.svg|80px]]<BR>A solid elliptic cylinder
|[[File:Cylinders.svg|120px]]<BR>A right and an oblique circular cylinder
|-
| [[Ellipsoid]]
| A surface that may be obtained from a [[sphere]] by deforming it by means of directional [[scaling (geometry)|scaling]]s, or more generally, of an [[affine transformation]]
|[[File:Ellipsoide.svg|150px]]<BR>Examples of ellipsoids
|<math>{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1:</math><br />''[[sphere]]'' (top, a=b=c=4),<br />
''[[spheroid]]'' (bottom left, a=b=5, c=3),<br />
''tri-axial'' ellipsoid (bottom right, a=4.5, b=6, c=3)]]
|-
| [[Lemon (geometry)|Lemon]]
| A [[lens (geometry)|lens]] (or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc)<ref name=mathworld>{{cite web|url=http://mathworld.wolfram.com/Lemon.html|title=Lemon|website=Wolfram [[:en:MathWorld|MathWorld]]|author=Weisstein, Eric W.|access-date=2019-11-04}}</ref>
|colspan=2|[[File:Lemon (geometry).png|90px]]
|-
| [[Hyperboloid]]
| A [[surface (mathematics)|surface]] that is generated by rotating a [[hyperbola]] around one of its [[Hyperbola#Nomenclature and features|principal axes]]
|colspan=2|[[File:Hyperboloid1.png|80px]]
|}

==Techniques==

Various techniques and tools are used in solid geometry. Among them, [[analytic geometry]] and [[Vector (geometric)|vector]] techniques have a major impact by allowing the systematic use of [[system of linear equations|linear equations]] and [[Matrix (mathematics)|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 (mathematics)|Surface]]

==Notes==
{{Reflist}}

==References==
* [[Robert Baldwin Hayward]] (1890) [https://archive.org/details/elementssolidge00haywgoog/page/n10/mode/2up ''The Elements of Solid Geometry''] via [[Internet Archive]]
* {{Cite book
 | first = A. P.
 | last = Kiselev
 | translator-first = Alexander
 | translator-last = Givental
 | title = Geometry
 | volume = Book II. Stereometry
 | publisher = Sumizdat
 | year = 2008
 
}}

{{Authority control}}

{{DEFAULTSORT:Solid Geometry}}
[[Category:Euclidean solid geometry|*]]
[[Category:Lists of shapes|Solid geometry]]