(Redirected from Equilateral square pyramid)
Square pyramid
Type Johnson
J92J1J2
Faces4 congruent triangles
1 square
Edges8
Vertices5
Vertex configuration4 (32.4)
(34)
Schläfli symbol( ) ∨ {4}
Symmetry groupC4v, [4], (*44)
Rotation groupC4, [4]+, (44)
Volume${\displaystyle V={\frac {l^{2}h}{3}}}$
Dual polyhedronself
Properties convex
Net
3D model of a square pyramid

In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has C4v symmetry. If all edge lengths are equal, it is an equilateral square pyramid, [1] the Johnson solid J1.

## General square pyramid

A possibly oblique square pyramid with base length l and perpendicular height h has volume:

${\displaystyle V={\frac {1}{3}}l^{2}h}$.

## Right square pyramid

In a right square pyramid, all the lateral edges have the same length, and the sides other than the base are congruent isosceles triangles.

A right square pyramid with base length l and height h has surface area and volume:

${\displaystyle A=l^{2}+l{\sqrt {l^{2}+4h^{2}}}}$,
${\displaystyle V={\frac {1}{3}}l^{2}h}$.

The lateral edge length is:

${\displaystyle {\sqrt {h^{2}+{{l^{2}} \over {2}}}}}$;

the slant height is:

${\displaystyle {\sqrt {h^{2}+{{l^{2}} \over {4}}}}}$.

The dihedral angles are:

• between the base and a side:
${\displaystyle \arctan \left({{2\,h} \over {l}}\right)}$;
• between two sides:
${\displaystyle \arccos \left({{-l^{2}} \over {l^{2}+4\,h^{2}}}\right)}$.

## Equilateral square pyramid, Johnson solid J1

If all edges have the same length, then the sides are equilateral triangles, and the pyramid is an equilateral square pyramid, Johnson solid J1.

The Johnson square pyramid can be characterized by a single edge length parameter l.

The height h (from the midpoint of the square to the apex), the surface area A (including all five faces), and the volume V of an equilateral square pyramid are:

${\displaystyle h={\frac {1}{\sqrt {2}}}l}$ ,
${\displaystyle A=\left(1+{\sqrt {3}}\right)l^{2}}$,
${\displaystyle V={\frac {\sqrt {2}}{6}}l^{3}}$.

The dihedral angles of an equilateral square pyramid are:

• between the base and a side:
${\displaystyle \arctan {{\big (}{\sqrt {2}}{\big )}}\approx 54.73561^{\circ }}$.
${\displaystyle \arccos \left({-{\frac {1}{3}}}\right)\approx 109.47122^{\circ }}$.

## Graph

A square pyramid can be represented by the wheel graph W5.

## Related polyhedra and honeycombs

Regular pyramids
Digonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal Decagonal...
Improper Regular Equilateral Isosceles
A regular octahedron can be considered a square bipyramid, i.e. two Johnson square pyramids connected base-to-base. The tetrakis hexahedron can be constructed from a cube with short square pyramids added to each face. Square frustum is a square pyramid with the apex truncated.

Square pyramids fill space with tetrahedra, truncated cubes, or cuboctahedra. [2]

### Dual polyhedron

The square pyramid is topologically a self-dual polyhedron. The dual's edge lengths are different due to the polar reciprocation.

Dual of square pyramid Net of dual

## References

1. ^ Franz Hocevar, Solid Geometry, 1903, p. 44
2. ^