Square pyramid

Square pyramid
Type	Johnson J92 – J1 – J2
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

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 C_4v symmetry.

Johnson solid (J1)

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.

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

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

Pyramids

Tetrahedron	Square pyramid	Pentagonal pyramid	Hexagonal pyramid

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

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

Topology

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

• 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)