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, quadrilat ...

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 adj ...

base. If the

apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *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 Slant can refer to: Bias *Bias or other non-objectivity in journalism, politics, academia or other fields Technical * Slant range, in telecommunications, the line-of-sight distance between two points which are not at the same level * Slant d ...

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 uni ...

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 othe ...

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 isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

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 A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to be ...

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 o ...

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triakis octahedron has edg ...

s, or cuboctahedra.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 In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...

. The dual's edge lengths are different due to the

polar reciprocation In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into it ...

References

External links

* *
Square Pyramid
-- Interactive Polyhedron Model

georgehart.com: The Encyclopedia of Polyhedra (

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphi ...

br>model
{{Johnson solids Johnson solids Prismatoid polyhedra Pyramids and bipyramids Self-dual polyhedra