Square pyramid

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Square pyramid
Square pyramid.png
Type Johnson
J92J1J2
Faces 4 triangles
1 square
Edges 8
Vertices 5
Vertex configuration 4(32.4)
(34)
Symmetry group C4v, [4], (*44)
Rotation group C4, [4]+, (44)
Dual polyhedron self
Properties convex
Net
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 will have C4v symmetry.

A Johnson solid is one of 92 strictly convex regular-faced polyhedra, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. They are named by Norman Johnson who first enumerated the set in 1966.

Johnson solid (J1)[edit]

If the sides are all equilateral triangles, the pyramid is one of the Johnson solids (J1). The 92 Johnson solids were named and described by Norman Johnson in 1966. galen verg...

The Johnson square pyramid can be characterized by a single edge-length parameter a. 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=\frac{1}{\sqrt{2}}a
A=(1+\sqrt{3})a^2
V=\frac{\sqrt{2}}{6}a^3.

Other square pyramids[edit]

Other square pyramids have isosceles triangle sides.

For square pyramids in general, with base length l and height h, the surface area and volume are:

A=l^2+l\sqrt{l^2+(2h)^2}
V=\frac{1}{3}l^2h.

Related polyhedra[edit]

Pyramids
Tetrahedron Square pyramid Pentagonal pyramid Hexagonal pyramid
Tetrahedron.svg Square pyramid.png Pentagonal pyramid.png Hexagonal 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.

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.png Dual square pyramid net.png

Topology[edit]

Like all pyramids, the square pyramid is self-dual, having the same number of vertices as faces.

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

External links[edit]