Pentagonal pyramid

Pentagonal pyramid
Type	Johnson J1 - J2 - J3
Faces	5 triangles 1 pentagon
Edges	10
Vertices	6
Vertex configuration	5(32.5) (35)
Symmetry group	C5v
Dual polyhedron	self
Properties	convex
net

In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the vertex). Like any pyramid, it is self-dual.

The regular pentagonal pyramid has a base that is a regular pentagon and lateral faces that are equilateral triangles. It is one of the Johnson solids (J₂). Its height H, from the midpoint of the pentagonal face to the apex, (as a function of a, where a is the side length), can be computed as:

Its surface area, A, can be computed as the area of pentagonal base plus five times the area of one triangle:

Its volume when an edge length is known can be figured out with this formula:

It can be seen as the "lid" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, J₁₁. The 92 Johnson solids were named and described by Norman Johnson in 1966.

More generally an order-2 vertex-uniform pentagonal pyramid can be defined with a regular pentagonal base and 5 isosceles triangle sides of any height.

External links

• Eric W. Weisstein, Pentagonal pyramid (Johnson solid) at MathWorld.
• Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra ( VRML model)

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