Square pyramid - wikipedia Square pyramid Square pyramid Type Johnson J -- J -- J Faces 4 triangles 1 square Edges 8 Vertices 5 Vertex configuration 4 ( 3. 4 ) ( 3 ) Schläfli symbol ( ) ∨ ( 4 ) Symmetry group C, ( 4 ), ( * 44 ) Rotation group C, ( 4 ), ( 44 ) Dual polyhedron self Properties convex Net In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has C symmetry. If all edges are equal, it is an equilateral square pyramid, the Johnson solid J. Contents 1 General square pyramid 2 Right square pyramid 3 Equilateral square pyramid, Johnson solid J 4 Graph 5 Related polyhedra and honeycombs 5.1 Dual polyhedron 6 References 7 External links General square pyramid A possibly oblique square pyramid with base length l and perpendicular height h has volume : V = 1 3 l 2 h ( displaystyle V = ( \frac ( 1 ) ( 3 ) ) l ^ ( 2 ) h ). Right square pyramid In a right square pyramid, all the lateral edges have the same length, and the sides other than the base are congruent isosceles triangles. A right square pyramid with base length l and height h has surface area and volume : A = l 2 + l l 2 + ( 2 h ) 2 ( displaystyle A = l ^ ( 2 ) + l ( sqrt ( l ^ ( 2 ) + ( 2h ) ^ ( 2 ) ) ) ) V = 1 3 l 2 h ( displaystyle V = ( \frac ( 1 ) ( 3 ) ) l ^ ( 2 ) h ). The lateral edge length is : h 2 + l 2 2 ( displaystyle ( sqrt ( h ^ ( 2 ) + ( ( l ^ ( 2 ) ) over ( 2 ) ) ) ) ), and the slant height is : h 2 + l 2 4 ( displaystyle ( sqrt ( h ^ ( 2 ) + ( ( l ^ ( 2 ) ) over ( 4 ) ) ) ) ). The dihedral angles are : between the base and a side : arctan ⁡ ( 2 h l ) ( displaystyle arctan left ( ( ( 2 , h ) over ( l ) ) \right ) ) ; between two sides : arccos ⁡ ( − l 2 l 2 + 4 h 2 ) ( displaystyle arccos left ( ( ( - l ^ ( 2 ) ) over ( l ^ ( 2 ) + 4 , h ^ ( 2 ) ) ) \right ) ). Equilateral square pyramid, Johnson solid J If all the edges have the same length, then the sides are equilateral triangles, and the pyramid is an equilateral square pyramid, Johnson solid J. The Johnson square pyramid can be characterized by a single edge - length parameter l. The height h ( from the midpoint of the square to the apex ), the surface area A ( including all five faces ), and the volume V of such a pyramid are : h = 1 2 l ( displaystyle h = ( \frac ( 1 ) ( sqrt ( 2 ) ) ) l ) A = ( 1 + 3 ) l 2 ( displaystyle A = ( 1 + ( sqrt ( 3 ) ) ) l ^ ( 2 ) ) V = 2 6 l 3. ( displaystyle V = ( \frac ( sqrt ( 2 ) ) ( 6 ) ) l ^ ( 3 ). ) Graph A square pyramid can be represented by the Wheel graph W. Related polyhedra and honeycombs Regular pyramids Digonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal... Improper Regular Equilateral Isosceles A regular octahedron can be considered a square bipyramid, i.e. two Johnson square pyramids connected base - to - base. The tetrakis hexahedron can be constructed from a cube with short square pyramids added to each face. Square frustum is a square pyramid with the apex truncated. Square pyramids fill space with tetrahedra, truncated cubes or cuboctahedra. Dual polyhedron The square pyramid is topologically a self - dual polyhedron. The dual edge lengths are different due to the polar reciprocation. Dual Square pyramid Net of dual References Jump up ^ Franz Hocevar, Solid Geometry, 1903, p. 44 Jump up ^ http://woodenpolyhedra.web.fc2.com/JohnsonHoneycomb.pdf External links Eric W. Weisstein, Square pyramid ( Johnson solid ) at MathWorld. Weisstein, Eric W. `` Wheel graph ''. MathWorld. Square Pyramid -- Interactive Polyhedron Model Virtual Reality Polyhedra www.georgehart.com : The Encyclopedia of Polyhedra ( VRML model ) 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 bifrustum rotunda birotunda Degenerate polyhedra are in italics. Johnson solids Prismatoids and rotundae square pyramid pentagonal pyramid triangular cupola square cupola pentagonal cupola pentagonal rotunda Modified pyramids and dipyramids elongated triangular pyramid elongated square pyramid elongated pentagonal pyramid gyroelongated square pyramid gyroelongated pentagonal pyramid triangular dipyramid square dipyramid pentagonal dipyramid elongated triangular dipyramid elongated square dipyramid elongated pentagonal dipyramid gyroelongated square dipyramid Modified cupolae and rotundae elongated triangular cupola elongated square cupola elongated pentagonal cupola elongated pentagonal rotunda gyroelongated triangular cupola gyroelongated square cupola gyroelongated pentagonal cupola gyroelongated pentagonal rotunda gyrobifastigium triangular orthobicupola square orthobicupola square gyrobicupola pentagonal orthobicupola pentagonal gyrobicupola pentagonal orthocupolarotunda pentagonal gyrocupolarotunda pentagonal orthobirotunda elongated triangular orthobicupola elongated triangular gyrobicupola elongated square gyrobicupola elongated pentagonal orthobicupola elongated pentagonal gyrobicupola elongated pentagonal orthocupolarotunda elongated pentagonal gyrocupolarotunda elongated pentagonal orthobirotunda elongated pentagonal gyrobirotunda gyroelongated triangular bicupola gyroelongated square bicupola gyroelongated pentagonal bicupola gyroelongated pentagonal cupolarotunda gyroelongated pentagonal birotunda Augmented prisms augmented triangular prism biaugmented triangular prism triaugmented triangular prism augmented pentagonal prism biaugmented pentagonal prism augmented hexagonal prism parabiaugmented hexagonal prism metabiaugmented hexagonal prism triaugmented hexagonal prism Modified Platonic solids augmented dodecahedron parabiaugmented dodecahedron metabiaugmented dodecahedron triaugmented dodecahedron metabidiminished icosahedron tridiminished icosahedron augmented tridiminished icosahedron Modified Archimedean solids augmented truncated tetrahedron augmented truncated cube biaugmented truncated cube augmented truncated dodecahedron parabiaugmented truncated dodecahedron metabiaugmented truncated dodecahedron triaugmented truncated dodecahedron gyrate rhombicosidodecahedron parabigyrate rhombicosidodecahedron metabigyrate rhombicosidodecahedron trigyrate rhombicosidodecahedron diminished rhombicosidodecahedron paragyrate diminished rhombicosidodecahedron metagyrate diminished rhombicosidodecahedron bigyrate diminished rhombicosidodecahedron parabidiminished rhombicosidodecahedron metabidiminished rhombicosidodecahedron gyrate bidiminished rhombicosidodecahedron tridiminished rhombicosidodecahedron Miscellaneous snub disphenoid snub square antiprism sphenocorona augmented sphenocorona sphenomegacorona hebesphenomegacorona disphenocingulum bilunabirotunda triangular hebesphenorotunda ( See also List of Johnson solids, a sortable table ) Retrieved from `` https://en.wikipedia.org/w/index.php?title=Square_pyramid&oldid=843300081 '' Categories : Self - dual polyhedra Prismatoid polyhedra Johnson solids Pyramids and bipyramids Talk Contents About Wikipedia Български Català Español Esperanto Euskara Français 한국어 Italiano Nederlands 日本 語 Norsk Português Română Русский Slovenščina ไทย Türkçe 中文 10 more Edit links This page was last edited on 28 May 2018, at 06 : 47. 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Question: how many vertices does a square based pyramid have
Answer: 5