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, quadrila ...

having a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

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 triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

s. 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 angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...

s 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 triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

s, 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 Johns ...

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 the ...

, truncated cubes, or

cuboctahedra A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a t ...

.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