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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, called a ''lateral face''. It is a conic solid with polygonal base. A pyramid with an ''n''-sided base will have vertices, faces, and 2''n'' edges. All pyramids are self-dual.
A right pyramid has its apex directly above the centroid of its base. Nonright pyramids are called oblique pyramids. A regular pyramid has a regular polygon base and is usually implied to be a ''right pyramid''.〔William F. Kern, James R Bland,''Solid Mensuration with proofs'', 1938, p.46〕 〔(Civil Engineers' Pocket Book: A Reference-book for Engineers )〕
When unspecified, a pyramid is usually assumed to be a ''regular'' square pyramid, like the physical pyramid structures. A triangle-based is more often called a tetrahedron.
Among oblique pyramids, like acute and obtuse triangles, a pyramid can be called ''acute'' if its apex above the interior of the base, and ''obtuse'' if its apex above the exterior of the base. A right-angled pyramid has its apex above an edge or vertex of the base. In a tetrahedron these qualifiers will change based on which face is considered the base.
Pyramids are a subclass of the prismatoids. Pyramids can be doubled into bipyramid by adding a second offset point on the other side of the base plane.
==Right pyramids with a regular base==
A right pyramid with a regular-based has isosceles triangle sides, with symmetry is C''n''v or (), with order 2''n''. It can be given an extended Schläfli symbol ( ) ∨ , representing a point, ( ), joined (orthogonally offset) to a regular polygon, . A join operation creates a new edge between all pairs of vertices of the two joined figures.
The trigonal or triangular pyramid with all equilateral triangles faces becomes the regular tetrahedron, one of the Platonic solids. A lower symmetry case of the triangular pyramid is C3v which has an equilateral triangle base, and 3 identical isosceles triangle sides. The square and pentagonal pyramids can also be composed of regular convex polygons, in which case they are Johnson solids.
If all edges of a square pyramid (or any convex polyhedron) are tangent to a sphere so that the average position of the tangential points are at the center of the sphere, then the pyramid is said to be canonical, and it forms half of a regular octahedron.

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