Square pyramid

The square pyramid, or squippy, is a pyramid with a square base and 4 triangles as sides. The version with equilateral triangles as sides is the first of the 92 Johnson solids (J1). In what follows, unless otherwise specified, this is what will be meant by a "square pyramid", even though other variants with isosceles triangles as sides exist.

Two square pyramids can be joined together at their square base to form a regular octahedron. As such it can also be thought of as a diminished octahedron.

In addition to being a point atop a square, the square pyramid has a second representation as a segmentohedron, as a dyad atop a triangle, obtained from removing a vertex from the octahedron.

Abstractly, the square pyramid is the simplest non-regular polytope without digonal faces overall.

It is one of three regular polygonal pyramids to be CRF. The others are the regular tetrahedron (triangular pyramid) and the pentagonal pyramid.

The orthoplecial pyramids, a family of Blind polytopes, are one way to generalize the square pyramid to any number of dimensions.

Vertex coordinates
Coordinates for a square pyramid of edge length 1 are given by:


 * $$\left(\pm\frac{\sqrt2}{2},\,0,\,0\right),$$
 * $$\left(0,\,\pm\frac{\sqrt2}{2},\,0\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt2}{2}\right).$$

Representations
A square pyramid has the following Coxeter diagrams:


 * ox4oo&#x (full symmetry)
 * ox ox&#x (rectangle pyramid)
 * oxx&#x (isosceles trapezoid pyramid)

General variant
For the general square pyramid with base edges of length b and lacing edges of length l, its height is given by $$\sqrt{l^2-\frac{b^2}{2}}$$, its circumradius by $$\frac{l}{2\sqrt{1-\frac{b^2}{2l^2}}}$$, and its volume is given by $$\frac13b^2\sqrt{l^2-\frac{b^2}{2}}$$.

A square pyramid with base edges of length 1 and side edges of length $$\sqrt2$$ appears as the vertex figure of the octahedral prism, and with side edges of length $$\sqrt3$$ as the vertex figure of the truncated hexadecachoron.

Variations
Various other tetragonal pyramids with less symmetry exist, differentiated by their bases. These include:


 * Rectangular pyramid - 1 rectangle and 2 pairs of isosceles tiranlges
 * Rhombic pyramid - 1 rhombus and 4 identical scalene triangles
 * Isosceles trapezoidal pyramid - 1 isosceles trapezoid, 2 separate isosceles triangles and 2 different scalene triangles
 * Kite pyramid - 1 kite and 2 pairs of scalene triangles
 * Irregular tetragonal pyramid - no symmetry

Related polyhedra
A cube can be attached to the base of a square pyramid to form the elongated square pyramid. If a square antiprism is attached instead, the result is the gyroelongated square pyramid.

The dihedral angle between two adjacent triangles is a supplementary angle to that of the regular tetrahedron. Thus, augmenting one of the triangular faces with a tetrahedron produces coplanar faces, disqualifying the resulting polyhedron from being a Johnson solid.