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Square pyramid
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a square pyramid is a pyramid
Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
 having a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.

he sides are all equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s, the pyramid is one of the Johnson solid
Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
s (J1). The 92 Johnson solids were named and described by Norman Johnson
Norman Johnson

Norman W. Johnson is a mathematician, previously at Wheaton College, Massachusetts, Norton, Massachusetts. He earned his Ph.D. from the University of Toronto in 1966 with a dissertation title of The Theory of Uniform Polytopes and Honeycombs under the supervision of H....
 in 1966.

The Johnson square pyramid can be characterized by a single edge-length parameter a.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a square pyramid is a pyramid
Pyramid (geometry)

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle....
 having a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.

Johnson solid (J1)

If the sides are all equilateral triangle
Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s, the pyramid is one of the Johnson solid
Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
s (J1). The 92 Johnson solids were named and described by Norman Johnson
Norman Johnson

Norman W. Johnson is a mathematician, previously at Wheaton College, Massachusetts, Norton, Massachusetts. He earned his Ph.D. from the University of Toronto in 1966 with a dissertation title of The Theory of Uniform Polytopes and Honeycombs under the supervision of H....
 in 1966.

The Johnson square pyramid can be characterized by a single edge-length parameter a. 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:

Other square pyramids


Other square pyramids have isosceles triangle sides. For example the Great Pyramid of Giza
Great Pyramid of Giza

The Great Pyramid of Giza, also called Khufu's Pyramid or the Pyramid of Khufu, and Pyramid of Cheops, is the oldest and largest of the three Egyptian pyramidss in the Giza Necropolis bordering what is now Cairo , Egypt, and is the only remaining member of the Seven Wonders of the Ancient World....
, has isosceles triangles of base 756 feet and slant height
Slant height

The slant height of a right circular cone is the distance from any point on the circle to the apex of the cone.The slant height of a cone is given by the formula , where is the radius of the circle and is the height from the center of the circle to the apex of the cone....
 719 feet. That pyramid has the interesting property that the slant height (along the bisector of a face) is very nearly equal to the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
 times the height, in which case the area of each triangular face is equal to the square of the pyramid's height.

For square pyramids in general, with base length l and height h, the surface area and volume are:

Related polyhedra


Tetrakishexahedron
A regular octahedron
Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
 can be considered a square bipyramid, with two Johnson square pyramids connected base-to-base.		The tetrakis hexahedron
Tetrakis hexahedron

A Conway kis operator hexahedron is an Archimedean solid solid, or a Catalan solid. Its dual is the truncated octahedron. It can be seen as a cube with square pyramids covering each square face....
 can be considered a cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
 with short square pyramids added to each face.


Topology


Like all pyramids, the square pyramid is self-dual, containing the same number of vertices and faces.

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

In the mathematical discipline of graph theory, a wheel graph Wn is a graph with n vertices, formed by connecting a single vertex to all vertices of an -Cycle graph....
 W5.

See also

  • Bipyramid
    Bipyramid

    An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal Pyramid and its mirror image base-to-base.The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the 2 pyramid halves....
     - A bipyramid is two pyramids connected base to base.


External links

  • -- Interactive Polyhedron Model
  • www.georgehart.com: The Encyclopedia of Polyhedra (VRML
    VRML

    VRML is a standard file format for representing 3-D computer graphics interactive vector graphics, designed particularly with the World Wide Web in mind....
     )