Square pyramid

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Square pyramid
Square pyramid.png
Faces4 triangles
1 square
VerticesTrryru Ru], (*44)
Vertex configuration4(32.4)
Rotation groupC4, [4]+, (44)
Dual polyhedronself
Square pyramid net.svg

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 C4v symmetry. If all edges are equal, it is an equilateral square pyramid,[1] the Johnson solid J1.

General square pyramid[edit]

A possibly oblique square pyramid with base length l and perpendicular height h has volume:


Right square pyramid[edit]

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:


The lateral edge length is:


and the slant height is:


The dihedral angles are:

between the base and a side: ;
between two sides: .

Equilateral square pyramid, Johnson solid J1 [edit]

If all the edges have the same length, then the sides are equilateral triangles, and the pyramid is an equilateral square pyramid, Johnson solid J1.

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:


A square pyramid can be represented by the Wheel graph W5.

Related polyhedra and honeycombs[edit]

Regular pyramids
Digonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal Decagonal...
Improper Regular Equilateral Isosceles
Biangular pyramid1.png Tetrahedron.svg Square pyramid.png Pentagonal pyramid.png Hexagonal pyramid.png Heptagonal pyramid1.png Octagonal pyramid1.png Enneagonal pyramid1.png Decagonal pyramid1.png
Spherical digonal pyramid.png Spherical trigonal pyramid.png Spherical square pyramid.png Spherical pentagonal pyramid.png Spherical hexagonal pyramid.png Spherical heptagonal pyramid.png Spherical octagonal pyramid.png Spherical enneagonal pyramid.png Spherical decagonal pyramid.png
Square bipyramid.png Tetrakishexahedron.jpg Usech kvadrat piramid.png
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.[2]

Dual polyhedron[edit]

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
Dual Square Pyramid New.png Dual Square Pyramid Net New.png


  1. ^ Franz Hocevar, Solid Geometry, 1903, p. 44
  2. ^ http://woodenpolyhedra.web.fc2.com/JohnsonHoneycomb.pdf

External links[edit]

  • 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)