geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a square pyramid is a

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...

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

General square pyramid

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

V=\frac 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: :

A=l^2+l\sqrt

, :

V=\frac l^2 h

. The lateral edge length is: :

\sqrt

; the slant height is: :

\sqrt

. The dihedral angles are: :*between the base and a side: :::

\arctan \left(\right)

; :*between two sides: :::

\arccos \left(\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 In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnso ...

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: :

h=\frac l

, :

A=\left(1+\sqrt\right)l^2

, :

V=\frac l^3

. The dihedral angles of an equilateral square pyramid are: :*between the base and a side: :::

\arctan\approx54.73561^\circ

. :*between two (adjacent) sides: :::

\arccos\left(\right)\approx109.47122^\circ

Graph

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

Related polyhedra and honeycombs

''Square pyramids'' fill space with

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

, truncated cubes, or

cuboctahedra A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, i ...

.凸正多角面体充填一覧表 / List of Regular polygon faced convex honeycomb
/ref>

Dual polyhedron

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

References

External links

* *
Square Pyramid
-- Interactive Polyhedron Model

georgehart.com: The Encyclopedia of Polyhedra ( VRMLbr>model
{{Johnson solids Johnson solids Prismatoid polyhedra Pyramids and bipyramids Self-dual polyhedra