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.

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.

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.

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
Triangular Square Pentagonal Hexagonal ...
Regular Equilateral Isosceles
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]