# Square pyramid (EntityTopic, 23)

A square pyramid is a special case of a pyramid where the base is a square.

The square pyramid with equilateral triangular sides is the first of the 92 Johnson solids, and is referred to as J1. It can be obtained from an octahedron by cutting it in half.

## Coordinates

The coordinates of a square pyramid with side 2 and a base centered on the origin are:

(±√2, 0, 0)
(0, ±√2, 0)
(0, 0, √2)

These coordinates are obtained from those of an octahedron by removing one fertex, since joining two square pyramids makes an octahedron.

## Equations

• Variables:
l ⇒ length of edges of the square pyramid
total edge length = 8l
surface area = (1 + √3) · l2
volume = √26 · l3
[!x,!y] ⇒ triangle
[!z] ⇒ square

## Incidence matrix

Dual: Self-dual

 # TXID Type Name Va Vb Ea Eb 3a 4a 0 Va = point ; sweep: apex 1 Vb = point ; lateral: vertices 2 Ea 1 1 = digon ; sweep: vertices 3 Eb 0 2 = digon ; lateral: edges 4 3a 1 2 2 1 = triangle ; sweep: edges 5 4a 0 4 0 4 = base of pyramid: square ; lateral: faces 6 C1a 1 4 4 4 4 1 = square pyramid ;

## Usage as facets

 Notable Trishapes Regular: tetrahedron • cube • octahedron • dodecahedron • icosahedron Direct truncates: tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate Mesotruncates: stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron Snubs: snub staurohedron • snub rhodohedron Curved: sphere • torus • cylinder • cone • frustum • crind

 8. 21Cone 9. 1Square pyramid 10. 111Triangular prism List of tapertopes