Square Pyramid
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Square Pyramid
Square pyramid
TypeJohnson
J92 - J1 - J2
Faces4 triangles
1 square
Edges8
Vertices5
Vertex configuration4(32.4)
(34)
Schläfli symbol( ) ? {4}
Symmetry groupC4v, [4], (*44)
Rotation groupC4, [4]+, (44)
Dual polyhedronself
Propertiesconvex
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 edges 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:

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

The lateral edge length is:

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

and 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 the 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:

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

The dihedral angles of an equilateral square pyramid are:

• Between the base and a side:
${\displaystyle \arctan {{\biggl (}{\sqrt {2}}{\biggl )}}\approx 54.73561^{\circ }.}$
${\displaystyle \arccos {\biggl (}{-1 \over 3}{\biggl )}\approx 109.47122^{\circ }.}$

## 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 edge lengths are different due to the polar reciprocation.

Dual Square pyramid Net of dual
The text passage provides detailed information about square pyramids, including their geometric properties, formulas for volume and surface area, and specific details about right and equilateral square pyramids. It also includes formulas and numerical values for various parameters and angles.
no_question

the passage does not contain any explicit mathematical questions or requests for calculations. it is purely informational, providing definitions, properties, and formulas related to square pyramids.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Square Pyramid
Get Square Pyramid essential facts below. View Videos or join the Square Pyramid discussion. Add Square Pyramid to your PopFlock.com topic list for future reference or share this resource on social media.
Square Pyramid
Square pyramid
TypeJohnson
J92 - J1 - J2
Faces4 triangles
1 square
Edges8
Vertices5
Vertex configuration4(32.4)
(34)
Schläfli symbol( ) ? {4}
Symmetry groupC4v, [4], (*44)
Rotation groupC4, [4]+, (44)
Dual polyhedronself
Propertiesconvex
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 edges 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:

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

The lateral edge length is:

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

and 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 the 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:

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

The dihedral angles of an equilateral square pyramid are:

• Between the base and a side:
${\displaystyle \arctan {{\biggl (}{\sqrt {2}}{\biggl )}}\approx 54.73561^{\circ }.}$
${\displaystyle \arccos {\biggl (}{-1 \over 3}{\biggl )}\approx 109.47122^{\circ }.}$

## 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 edge lengths are different due to the polar reciprocation.

Dual Square pyramid Net of dual
user
The text passage provides detailed information about square pyramids, including their geometric properties, formulas for volume and surface area, and specific details about right and equilateral square pyramids. It also includes formulas and numerical values for various parameters and angles.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or requests for calculations. It is purely informational, providing definitions, properties, and formulas related to square pyramids.