Square pyramid | |
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Type |
Johnson J_{92} – J_{1} – J_{2} |
Faces | 4
congruent
triangles 1 square |
Edges | 8 |
Vertices | 5 |
Vertex configuration | 4 (3^{2}.4) (3^{4}) |
Schläfli symbol | ( ) ∨ {4} |
Symmetry group | C_{4v}, [4], (*44) |
Rotation group | C_{4}, [4]^{+}, (44) |
Volume | |
Dual polyhedron | self |
Properties | convex |
Net | |
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 C_{4v} symmetry. If all edge lengths are equal, it is an equilateral square pyramid,^{ [1]} the Johnson solid J_{1}.
A possibly oblique square pyramid with base length l and perpendicular height h has volume:
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:
The lateral edge length is:
the slant height is:
The dihedral angles are:
If all edges have the same length, then the sides are equilateral triangles, and the pyramid is an equilateral square pyramid, Johnson solid J_{1}.
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:
The dihedral angles of an equilateral square pyramid are:
A square pyramid can be represented by the wheel graph W_{5}.
Regular pyramids | ||||||||
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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]}
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 |
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