“Square pyramid”的意思、由来-开放百科全书

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Square pyramid

释义

General square pyramid
Right square pyramid
Equilateral square pyramid, Johnson solid J₁ {{anchor|equilateral square pyramid}}
Graph
Related polyhedra and honeycombs
Dual polyhedron
References
External links

{{Infobox polyhedron
|image=Square pyramid.png
|type=Johnson
J₉₂ – J₁ – J₂
|faces=4 triangles
1 square
|edges=8
|vertices=5
|schläfli=( ) ∨ {4}
|symmetry=C_4v, [4], (*44)
|rotation_group=C₄, [4]⁺, (44)
|vertex_config=4(3².4)
(3⁴)
|volume=
|dual=self
|properties=convex
|net=Square pyramid net.svg
}}

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 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:

.

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:

.

The lateral edge length is:

,

and the slant height is:

.

The dihedral angles are:

between the base and a side: ;

between two sides: .

Equilateral square pyramid, Johnson solid J₁ {{anchor|equilateral square pyramid}}

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 such a pyramid are:

Graph

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

Related polyhedra and honeycombs

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

References

1. ^Franz Hocevar, Solid Geometry, 1903, [https://books.google.com/books?id=0OAXAAAAYAAJ&pg=PA44 p. 44]
2. ^http://woodenpolyhedra.web.fc2.com/JohnsonHoneycomb.pdf

External links

{{Mathworld2 | urlname = SquarePyramid | title = Square pyramid | urlname2 = JohnsonSolid | title2 = Johnson solid }}
{{Mathworld | urlname = WheelGraph | title = Wheel graph}}
[https://web.archive.org/web/20071008222854/http://polyhedra.org/poly/show/45/square_pyramid Square Pyramid] -- Interactive Polyhedron Model
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra (VRML model)

4 : Self-dual polyhedra|Prismatoid polyhedra|Johnson solids|Pyramids and bipyramids

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