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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:

$H=a\sqrt{\frac{1}{2}-\sqrt{\frac{1}{20}}}$

, while its surface area, A, can be computed as:

$A=t^2(\sqrt{\frac{25}{16} + \sqrt{\frac{125}{64}}} + \sqrt{\frac{75}{16}})$

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.

The volume of a pentagonal pyramid is:

$V= s^3\sqrt{\frac{25}{144} + \sqrt{\frac{125}{5184}}}$