Set of pyramids
Faces	n triangles, 1 n-gon
Edges	2n
Vertices	n+1
Symmetry group	Cnv
Dual polyhedron	Self-duals
Properties	convex

This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation).

An n-sided pyramid is a polyhedron formed by connecting an n-sided polygonal base and a point, called the apex, by n triangular faces (n ≥ 3). In other words, it is a conic solid with polygonal base. Image File history File links Pyramid. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... Look up polygon in Wiktionary, the free dictionary. ... The symmetry group of an object (e. ... The symmetry group of an object (e. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ... // See Pyramid (geometry) A pyramid is a three-dimensional polyhedron formed by connecting an n-sided polygonal base and a point, called the apex, by n triangular faces (n ≥ 3). ... A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ... Look up polygon in Wiktionary, the free dictionary. ... In geometry, an apex is a descriptive label for a visual singular highest or most distant point or vertex in an isosceles triangle, pyramid or cone, usually contrasting with the opposite side called the base. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. ... A conic solid is the locus of all line segments between a simply connected region of a plane (the base) and a point (the apex) outside the plane. ...

When unspecified, the base is usually assumed to be square. For a triangular pyramid each face can serve as base, with the opposite vertex as apex. The regular tetrahedron, one of the Platonic solids, is a triangular pyramid. The square and pentagonal pyramids can also be constructed with all faces regular, and are therefore Johnson solids. All pyramids are self-dual. Look up vertex in Wiktionary, the free dictionary. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, a Platonic solid is a convex regular polyhedron. ... The elongated square gyrobicupola (J37), a Johnson solid In geometry, a Johnson solid is a convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. ...

Pyramids are a subclass of the prismatoids. The 1-skeleton of pyramid is a wheel graph. A prismatoid is a polyhedron where all vertices lie in two parallel planes. ... In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex, or CW complex, refers to the subspace Xn that is the union of the simplices of X (resp. ... In the mathematical discipline of graph theory a wheel graph is a special graph which can be visualized as a wheel. ...

Volume

The volume of a pyramid is where A is the area of the base and h the height from the base to the apex. This works for any location of the apex, provided that h is measured as the perpendicular distance from the plane which contains the base. The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ... Fig. ... In mathematics, a plane is the fundamental two-dimensional object. ...

This can be proven using calculus:

Assume that the dimensions of a cross section parallel to the base decrease linearly from the base to the apex. Then, the cross section at any height y is the base scaled by a factor of , where h is the height from the base to the apex. Since the area of any shape is multiplied by the square of the shape's scaling factor, the area of a cross section at height y is .

The volume is given by the integral

Surface Area

The surface area of a regular pyramid is where A_b is the area of the base, p is the perimeter of the base, and s is the slant height along the bisector of a face (ie the length from the midpoint of any edge of the base to the apex). Area is a physical quantity expressing the size of a part of a surface. ...

Pyramids with regular polygon faces

If all faces are regular polygons, the pyramid base can be a regular polygon of 3-, 4- or 5-sided:

The geometric center of a square-based pyramid is located on the symmetry axis, one quarter of the way from the base to the apex. A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the square pyramid, a pyramid with a square base and equilateral sides, is one of the Johnson solids (J1). ... In geometry, the Pentagonal pyramid is one of the Johnson solids (J2). ... Image File history File links Download high-resolution version (643x607, 26 KB)Tetrahedron, made by me using POV-Ray, see image:poly. ... Image File history File links Johnson solid #1 Another attempt at using user:Cyps image:Poly. ... Image File history File links Pentagonal_pyramid. ... In geometry, a Platonic solid is a convex regular polyhedron. ... The elongated square gyrobicupola (J37), a Johnson solid In geometry, a Johnson solid is a convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. ... The elongated square gyrobicupola (J37), a Johnson solid In geometry, a Johnson solid is a convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... For other uses, see Square. ... Look up pentagon in Wiktionary, the free dictionary. ...

Symmetry

If the base is regular and the apex is above the center, the symmetry group of the n-sided pyramid is C_nv of order 2n, except in the case of a regular tetrahedron, which has the larger symmetry group T_d of order 24, which has four versions of C_3v as subgroups. The rotation group is C_n of order n, except in the case of a regular tetrahedron, which has the larger rotation group T of order 12, which has four versions of C₃ as subgroups. The symmetry group of an object (e. ... In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. ...

See also

• Bipyramid
• Cone (geometry)
• Trigonal pyramid (chemistry)

A bipyramid is a polyhedron formed by joining two identical pyramids base-to-base. ... This article is about the geometric object, for other uses see Cone. ... In chemistry, a trigonal pyramid is a molecular geometry with one atom at the apex and three atoms at the corners of a trigonal base. ...

External links

• Eric W. Weisstein, Pyramid at MathWorld.
• Olshevsky, George, Pyramid at Glossary for Hyperspace.
• The Uniform Polyhedra

• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • VRML models (George Hart) <3> <4> <5>
• Paper models of pyramids