Cube - wikipedia Cube This article is about the geometric shape. For other uses, see Cube ( disambiguation ). Regular hexahedron ( Click here for rotating model ) Type Platonic solid Elements F = 6, E = 12 V = 8 ( χ = 2 ) Faces by sides 6 ( 4 ) Conway notation Schläfli symbols ( 4, 3 ) t ( 2, 4 ) or ( 4 ) × ( ) tr ( 2, 2 ) or ( ) × ( ) × ( ) Face configuration V3. 3.3. 3 Wythoff symbol 3 2 4 Coxeter diagram Symmetry O, B, ( 4, 3 ), ( * 432 ) Rotation group O, ( 4, 3 ), ( 432 ) References U, C, W Properties regular, convex zonohedron Dihedral angle 90 ° 4.4. 4 ( Vertex figure ) Octahedron ( dual polyhedron ) Net Net of cube In geometry, a cube is a three - dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. Contents 1 Orthogonal projections 2 Spherical tiling 3 Cartesian coordinates 4 Equation in R 5 Formulas 5.1 Point in space 6 Doubling the cube 7 Uniform colorings and symmetry 8 Geometric relations 9 Other dimensions 10 Related polyhedra 10.1 In uniform honeycombs and polychora 11 Cubical graph 12 See also 13 References 14 External links Orthogonal projections The cube has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A and B Coxeter planes. Orthogonal projections Centered by Face Vertex Coxeter planes Projective symmetry ( 4 ) ( 6 ) Tilted views Spherical tiling The cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. Orthographic projection Stereographic projection Cartesian coordinates For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are ( ± 1, ± 1, ± 1 ) while the interior consists of all points ( x, x, x ) with − 1 < x < 1. Equation in R In analytic geometry, a cube 's surface with center ( x, y, z ) and edge length of 2a is the locus of all points ( x, y, z ) such that max ( x − x 0, y − y 0, z − z 0 ) = a. ( displaystyle max ( x-x_ ( 0 ), y - y_ ( 0 ), z - z_ ( 0 ) ) = a. ) Formulas Cube ( Matemateca Ime - Usp ) Play media Revolution of a cube ( at Matemateca_IME - USP ) For a cube of edge length a ( displaystyle a ) : surface area 6 a 2 ( displaystyle 6a ^ ( 2 ) , ) volume a 3 ( displaystyle a ^ ( 3 ) , ) face diagonal 2 a ( displaystyle ( sqrt ( 2 ) ) a ) space diagonal 3 a ( \textstyle ( sqrt ( 3 ) ) a ) radius of circumscribed sphere 3 2 a ( displaystyle ( \frac ( sqrt ( 3 ) ) ( 2 ) ) a ) radius of sphere tangent to edges a 2 ( displaystyle ( \frac ( a ) ( sqrt ( 2 ) ) ) ) radius of inscribed sphere a 2 ( displaystyle ( \frac ( a ) ( 2 ) ) ) angles between faces ( in radians ) π 2 ( displaystyle ( \frac ( pi ) ( 2 ) ) ) As the volume of a cube is the third power of its sides a × a × a ( displaystyle a \times a \times a ), third powers are called cubes, by analogy with squares and second powers. A cube has the largest volume among cuboids ( rectangular boxes ) with a given surface area. Also, a cube has the largest volume among cuboids with the same total linear size ( length + width + height ). Point in space For a cube whose circumscribing sphere has radius R, and for a given point in its 3 - dimensional space with distances d from the cube 's eight vertices, we have : ∑ i = 1 8 d i 4 8 + 16 R 4 9 = ( ∑ i = 1 8 d i 2 8 + 2 R 2 3 ) 2. ( displaystyle ( \frac ( sum _ ( i = 1 ) ^ ( 8 ) d_ ( i ) ^ ( 4 ) ) ( 8 ) ) + ( \frac ( 16R ^ ( 4 ) ) ( 9 ) ) = left ( ( \frac ( sum _ ( i = 1 ) ^ ( 8 ) d_ ( i ) ^ ( 2 ) ) ( 8 ) ) + ( \frac ( 2R ^ ( 2 ) ) ( 3 ) ) \right ) ^ ( 2 ). ) Doubling the cube Doubling the cube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number. Uniform colorings and symmetry Octahedral symmetry tree The cube has three uniform colorings, named by the colors of the square faces around each vertex : 111, 112, 123. The cube has three classes of symmetry, which can be represented by vertex - transitive coloring the faces. The highest octahedral symmetry O has all the faces the same color. The dihedral symmetry D comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D is also a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol. Name Regular hexahedron Square prism Rectangular cuboid Rhombic prism Trigonal trapezohedron Coxeter diagram Schläfli symbol ( 4, 3 ) ( 4 ) × ( ) rr ( 4, 2 ) s ( 2, 4 ) ( ) tr ( 2, 2 ) ( ) × 2 ( ) Wythoff symbol 3 4 2 4 2 2 2 2 2 Symmetry O ( 4, 3 ) ( * 432 ) ( 4, 2 ) ( * 422 ) ( 4, 2 ) ( 2 * 2 ) ( 2, 2 ) ( * 222 ) ( 6, 2 ) ( 2 * 3 ) Symmetry order 24 16 8 8 12 Image ( uniform coloring ) ( 111 ) ( 112 ) ( 112 ) ( 123 ) ( 112 ) ( 111 ), ( 112 ) Geometric relations The 11 nets of the cube. These familiar six - sided dice are cube - shaped. A cube has eleven nets ( one shown above ) : that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the same color, one would need at least three colors. The cube is the cell of the only regular tiling of three - dimensional Euclidean space. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron ( every face has point symmetry ). The cube can be cut into six identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained ( with pairs of coplanar triangles combined into rhombic faces ). Other dimensions The analogue of a cube in four - dimensional Euclidean space has a special name -- a tesseract or hypercube. More properly, a hypercube ( or n - dimensional cube or simply n - cube ) is the analogue of the cube in n - dimensional Euclidean space and a tesseract is the order - 4 hypercube. A hypercube is also called a measure polytope. There are analogues of the cube in lower dimensions too : a point in dimension 0, a segment in one dimension and a square in two dimensions. Related polyhedra The dual of a cube is an octahedron. The hemicube is the 2 - to - 1 quotient of the cube. The quotient of the cube by the antipodal map yields a projective polyhedron, the hemicube. If the original cube has edge length 1, its dual polyhedron ( an octahedron ) has edge length 2 / 2 ( displaystyle scriptstyle ( sqrt ( 2 ) ) / 2 ). The cube is a special case in various classes of general polyhedra : Name Equal edge - lengths ? Equal angles ? Right angles ? Cube Yes Yes Yes Rhombohedron Yes Yes No Cuboid No Yes Yes Parallelepiped No Yes No quadrilaterally faced hexahedron No No No The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron ; more generally this is referred to as a demicube. These two together form a regular compound, the stella octangula. The intersection of the two forms a regular octahedron. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself ; the other symmetries of the cube map the two to each other. One such regular tetrahedron has a volume of 1 / 3 of that of the cube. The remaining space consists of four equal irregular tetrahedra with a volume of 1 / 6 of that of the cube, each. The rectified cube is the cuboctahedron. If smaller corners are cut off we get a polyhedron with six octagonal faces and eight triangular ones. In particular we can get regular octagons ( truncated cube ). The rhombicuboctahedron is obtained by cutting off both corners and edges to the correct amount. A cube can be inscribed in a dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron 's faces ; taking all such cubes gives rise to the regular compound of five cubes. If two opposite corners of a cube are truncated at the depth of the three vertices directly connected to them, an irregular octahedron is obtained. Eight of these irregular octahedra can be attached to the triangular faces of a regular octahedron to obtain the cuboctahedron. The cube is topologically related to a series of spherical polyhedra and tilings with order - 3 vertex figures. * n32 symmetry mutation of regular tilings : ( n, 3 ) Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic ( 2, 3 ) ( 3, 3 ) ( 4, 3 ) ( 5, 3 ) ( 6, 3 ) ( 7, 3 ) ( 8, 3 ) ( ∞, 3 ) ( 12i, 3 ) ( 9i, 3 ) ( 6i, 3 ) ( 3i, 3 ) The cuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron. Uniform octahedral polyhedra Symmetry : ( 4, 3 ), ( * 432 ) ( 4, 3 ) ( 432 ) ( 1, 4, 3 ) = ( 3, 3 ) ( * 332 ) ( 3, 4 ) ( 3 * 2 ) ( 4, 3 ) t ( 4, 3 ) r ( 4, 3 ) r ( 3 ) t ( 3, 4 ) t ( 3 ) ( 3, 4 ) ( 3 ) rr ( 4, 3 ) s ( 3, 4 ) tr ( 4, 3 ) sr ( 4, 3 ) h ( 4, 3 ) ( 3, 3 ) h ( 4, 3 ) t ( 3, 3 ) s ( 3, 4 ) s ( 3 ) = = = = or = or = Duals to uniform polyhedra V4 V3. 8 V ( 3.4 ) V4. 6 V3 V3. 4 V4. 6.8 V3. 4 V3 V3. 6 V3 The cube is topologically related as a part of sequence of regular tilings, extending into the hyperbolic plane : ( 4, p ), p = 3, 4, 5... * n42 symmetry mutation of regular tilings : ( 4, n ) Spherical Euclidean Compact hyperbolic Paracompact ( 4, 3 ) ( 4, 4 ) ( 4, 5 ) ( 4, 6 ) ( 4, 7 ) ( 4, 8 )... ( 4, ∞ ) With dihedral symmetry, Dih, the cube is topologically related in a series of uniform polyhedra and tilings 4.2 n. 2n, extending into the hyperbolic plane : * n42 symmetry mutation of truncated tilings : 4.2n. 2n Symmetry * n42 ( n, 4 ) Spherical Euclidean Compact hyperbolic Paracomp. * 242 ( 2, 4 ) * 342 ( 3, 4 ) * 442 ( 4, 4 ) * 542 ( 5, 4 ) * 642 ( 6, 4 ) * 742 ( 7, 4 ) * 842 ( 8, 4 )... * ∞ 42 ( ∞, 4 ) Truncated figures Config. 4.4. 4 4.6. 6 4.8. 8 4.10. 10 4.12. 12 4.14. 14 4.16. 16 4. ∞. ∞ n - kis figures Config. V4. 4.4 V4. 6.6 V4. 8.8 V4. 10.10 V4. 12.12 V4. 14.14 V4. 16.16 V4. ∞. ∞ All these figures have octahedral symmetry. The cube is a part of a sequence of rhombic polyhedra and tilings with ( n, 3 ) Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. Symmetry mutations of dual quasiregular tilings : V ( 3. n ) * n32 Spherical Euclidean Hyperbolic * 332 * 432 * 532 * 632 * 732 * 832... * ∞ 32 Tiling Conf. V ( 3.3 ) V ( 3.4 ) V ( 3.5 ) V ( 3.6 ) V ( 3.7 ) V ( 3.8 ) V ( 3. ∞ ) The cube is a square prism : Family of uniform prisms Polyhedron Coxeter Tiling Config. 3.4. 4 4.4. 4 5.4. 4 6.4. 4 7.4. 4 8.4. 4 9.4. 4 10.4. 4 11.4. 4 12.4. 4 As a trigonal trapezohedron, the cube is related to the hexagonal dihedral symmetry family. Uniform hexagonal dihedral spherical polyhedra Symmetry : ( 6, 2 ), ( * 622 ) ( 6, 2 ), ( 622 ) ( 6, 2 ), ( 2 * 3 ) ( 6, 2 ) t ( 6, 2 ) r ( 6, 2 ) t ( 2, 6 ) ( 2, 6 ) rr ( 6, 2 ) tr ( 6, 2 ) sr ( 6, 2 ) s ( 2, 6 ) Duals to uniforms V6 V12 V6 V4. 4.6 V2 V4. 4.6 V4. 4.12 V3. 3.3. 6 V3. 3.3. 3 Regular and uniform compounds of cubes Compound of three cubes Compound of five cubes In uniform honeycombs and polychora It is an element of 9 of 28 convex uniform honeycombs : Cubic honeycomb Truncated square prismatic honeycomb Snub square prismatic honeycomb Elongated triangular prismatic honeycomb Gyroelongated triangular prismatic honeycomb Cantellated cubic honeycomb Cantitruncated cubic honeycomb Runcitruncated cubic honeycomb Runcinated alternated cubic honeycomb It is also an element of five four - dimensional uniform polychora : Tesseract Cantellated 16 - cell Runcinated tesseract Cantitruncated 16 - cell Runcitruncated 16 - cell Cubical graph Cubical graph Named after Q Vertices 8 Edges 12 Radius Diameter Girth Automorphisms 48 Chromatic number Properties Hamiltonian, regular, symmetric, distance - regular, distance - transitive, 3 - vertex - connected, planar graph The skeleton of the cube ( the vertices and edges ) form a graph, with 8 vertices, and 12 edges. It is a special case of the hypercube graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid. An extension is the three dimensional k - ary Hamming graph, which for k = 2 is the cube graph. Graphs of this sort occur in the theory of parallel processing in computers. See also Tesseract Trapezohedron Miscellaneous cubes Cube ( film ) Diamond cubic Lövheim cube of emotion Cube of Heymans Necker Cube OLAP cube Prince Rupert 's cube Rubik 's Cube The Cube ( game show ) Unit cube Yoshimoto Cube Kaaba References Jump up ^ English cube from Old French < Latin cubus < Greek κύβος ( kubos ) meaning `` a cube, a die, vertebra ''. In turn from PIE * keu ( b ) -, `` to bend, turn ''. Jump up ^ Park, Poo - Sung. `` Regular polytope distances '', Forum Geometricorum 16, 2016, 227 - 232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf Jump up ^ Weisstein, Eric W. `` Cube ''. MathWorld. Jump up ^ Weisstein, Eric W. `` Cubical graph ''. MathWorld. External links Weisstein, Eric W. `` Cube ''. MathWorld. Cube : Interactive Polyhedron Model * Volume of a cube, with interactive animation Cube ( Robert Webb 's site ) Convex polyhedra Platonic solids ( regular ) tetrahedron cube octahedron dodecahedron icosahedron Archimedean solids ( semiregular or uniform ) truncated tetrahedron cuboctahedron truncated cube truncated octahedron rhombicuboctahedron truncated cuboctahedron snub cube icosidodecahedron truncated dodecahedron truncated icosahedron rhombicosidodecahedron truncated icosidodecahedron snub dodecahedron Catalan solids ( duals of Archimedean ) triakis tetrahedron rhombic dodecahedron triakis octahedron tetrakis hexahedron deltoidal icositetrahedron disdyakis dodecahedron pentagonal icositetrahedron rhombic triacontahedron triakis icosahedron pentakis dodecahedron deltoidal hexecontahedron disdyakis triacontahedron pentagonal hexecontahedron Dihedral regular dihedron hosohedron Dihedral uniform prisms antiprisms duals : bipyramids trapezohedra Dihedral others pyramids truncated trapezohedra gyroelongated bipyramid cupola bicupola pyramidal frusta rotunda birotunda Degenerate polyhedra are in italics. Fundamental convex regular and uniform polytopes in dimensions 2 -- 10 Family I ( p ) / D E / E / E / E / E / F / G Regular polygon Triangle Square p - gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron Icosahedron Uniform 4 - polytope 5 - cell 16 - cell Tesseract Demitesseract 24 - cell 120 - cell 600 - cell Uniform 5 - polytope 5 - simplex 5 - orthoplex 5 - cube 5 - demicube Uniform 6 - polytope 6 - simplex 6 - orthoplex 6 - cube 6 - demicube 1 2 Uniform 7 - polytope 7 - simplex 7 - orthoplex 7 - cube 7 - demicube 1 2 3 Uniform 8 - polytope 8 - simplex 8 - orthoplex 8 - cube 8 - demicube 1 2 4 Uniform 9 - polytope 9 - simplex 9 - orthoplex 9 - cube 9 - demicube Uniform 10 - polytope 10 - simplex 10 - orthoplex 10 - cube 10 - demicube Uniform n - polytope n - simplex n - orthoplex n - cube n - demicube 1 2 k n - pentagonal polytope Topics : Polytope families Regular polytope List of regular polytopes and compounds Retrieved from `` https://en.wikipedia.org/w/index.php?title=Cube&oldid=804104886 '' Categories : Platonic solids Prismatoid polyhedra Space - filling polyhedra Volume Zonohedra Elementary shapes Cubes Hidden categories : Pages using deprecated image syntax Talk Contents About Wikipedia Afrikaans Aragonés Asturianu Azərbaycanca Bân - lâm - gú Беларуская Български Bosanski Català Чӑвашла Čeština Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Gaeilge Galego 贛 語 ગુજરાતી 한국어 Հայերեն हिन्दी Hrvatski Ido Bahasa Indonesia Italiano עברית Basa Jawa ಕನ್ನಡ ქართული Қазақша Kiswahili Кыргызча Latina Latviešu Lëtzebuergesch Lietuvių Magyar Македонски മലയാളം मराठी Bahasa Melayu Nederlands 日本 語 Nordfriisk Norsk Norsk nynorsk Occitan Олык марий Oʻzbekcha / ўзбекча ਪੰਜਾਬੀ پښتو Patois Piemontèis Plattdüütsch Polski Português Română Runa Simi Русский Scots Sicilianu Simple English Slovenčina Slovenščina Soomaaliga کوردی Српски / srpski Srpskohrvatski / српскохрватски Basa Sunda Suomi Svenska தமிழ் తెలుగు ไทย Türkçe Українська اردو Tiếng Việt 粵語 中文 Edit links This page was last edited on 6 October 2017, at 19 : 31. 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Question: how many vertices edges and faces does a cube have
Answer: It has 6 faces, 12 edges, and 8 vertices.