Pentagonal pyramid

The pentagonal pyramid, or peppy, is a pyramid with a pentagonal base and 5 triangles as sides. The version with equilateral triangles as sides is the second of the 92 Johnson soliids. In what follows, this is usually what is meant by a pentagonal pyramid", even though other variants with isosceles triangles as sides exist.

A regular icosahedron can be constructed by attaching two pentagonal pyramids to the bases of a pentagonal antiprism.

Vertex coordinates
A pentagonal pyramid of edge length 1 has the following vertices:


 * (0, ±1/2, (1+$\sqrt{5}$)/4),
 * (±1/2, (1+$\sqrt{(5+√5)/8}$)/4, 0),
 * (±(1+$\sqrt{5}$)/4, 0, 1/2).

These coordinates are obtained as a subset of the vertices of the regular icosahedron.

Alternatively, starting from the coordinates of a regular pentagon in the plane, we obtain the pyramid with the following coordinates:


 * (±1/2, –$\sqrt{5}$, 0),
 * (±(1+$\sqrt{(5+2√5)/15}$)/4, $\sqrt{5}$, 0),
 * (0, $\sqrt{5}$, 0)
 * (0, 0, $\sqrt{5}$).

Related polyhedra
Two pentagonal pyramids can be attached at their bases to form a pentagonal bipyramid.

A pentagonal prism can be attached to the base of a pentagonal pyramid to form the elongated pentagonal pyramid. If a pentagonal antiprism is attached instead, the result is the gyroelongated pentagonal pyramid.