Pyramid (geometry)

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Set of pyramids
Square Pyramid
Faces n triangles,
1 n-gon
Edges 2n
Vertices n+1
Symmetry group Cnv
Dual polyhedron Self-dual
Properties convex
The 1-skeleton of pyramid is a wheel graph
This article is about the polyhedron pyramid (a 3-dimensional shape); for other versions including architectural Pyramids, see Pyramid (disambiguation).

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. It is a conic solid with polygonal base.

An n-sided pyramid will have n+1 vertices, n+1 faces, and 2n edges. All pyramids are self-dual.

When unspecified, the base is usually assumed to be square.

If the base is a regular polygon and the apex is above the center the polygon, an n-gonal pyramid will has Cnv symmetry.

Pyramids are a subclass of the prismatoids.

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[edit] Pyramids with regular polygon faces

The regular tetrahedron, one of the Platonic solids, is a triangular pyramid all of whose faces are equilateral triangles. Besides the triangular pyramid, only the square and pentagonal pyramids can be composed of equilateral triangles, and in that case they are Johnson solids.

Tetrahedron Square pyramid Pentagonal pyramid

[edit] Star pyramids

Pyramids with regular star polygon bases can also be constructed.

For example the pentagrammic pyramid has a pentagram base and 5 intersecting equilateral triangle sides.

[edit] Volume

The volume of a pyramid is V = \frac{1}{3} Bh where B 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.

This can be proven using calculus:

It can be proved using similarity that the dimensions of a cross section parallel to the base increase linearly from the apex to the base. Then, the cross section at any height y is the base scaled by a factor of \frac{h-y}{h}, 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 \frac{A}{h^2}(h-y)^2.
The volume is given by the integral \frac{A}{h^2} \int_0^h (h-y)^2 \, dy = \frac{-A}{3h^2} (h-y)^3 \bigg|_0^h = \frac{1}{3}Ah.

(Trivially, the volume of a square-based pyramid with an apex half the height of its base can be seen to correspond to one sixth of a cube formed by fitting six such pyramids (in opposite pairs) about a center. Since the "base times height" then corresponds to one half of the cube's volume it is therefore three times the volume of the pyramid and the factor of one-third follows.)

[edit] Surface area

The surface area of a regular pyramid is A = A_b + \frac{ps}{2} where Ab 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).

[edit] See also

[edit] External links

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