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 (J2). In what follows, unless otherwise specified, this what will be meant by a "pentagonal pyramid", even though other variants with isosceles triangles as sides exist.

It is the vertex-first cap of the icosahedron. 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:


 * $$\left(0,\,±\frac12,\,\frac{1+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,\frac{1+\sqrt5}{4},\,0\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,0,\,\frac12\right).$$

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:


 * $$\left(±\frac{1}{2},\, -\sqrt{\frac{5+2\sqrt{5}}{20}},\,0\right),$$
 * $$\left(±\frac{1+\sqrt{5}}{4},\, \sqrt{\frac{5-\sqrt{5}}{40}},\,0\right),$$
 * $$\left(0,\, \sqrt{\frac{5+\sqrt{5}}{10}},\,0\right),$$
 * $$\left(0,\,0,\,\sqrt{\frac{5-\sqrt5}{10}}\right).$$

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.

General variant
For the general pentagonal pyramid with base edges of length b and lacing edges of length l, its height is given by $$\sqrt{l^2-b^2\frac{5+\sqrt5}{10}}$$, its circumradius by $$\frac{l}{2\sqrt{1-\frac{(5+\sqrt5)b^2}{10l^2}}}$$, and its volume is given by $$\frac{\sqrt{25+10\sqrt5}}{12}b^2\sqrt{l^2-b^2\frac{5+\sqrt5}{10}}$$.

Pentagonal pyramids occur as vertex figures of 6 uniform polychora, including the convex truncated hexacosichoron, truncated great hecatonicosachoron, truncated great faceted hexacosichoron, quasitruncated great stellated hecatonicosachoron, icosahedral prism, and small stellated dodecahedral prism.