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Pyramid (geometry)

Star pyramids

Pyramids with regular star polygon bases are called '''star pyramids'''. For example, the '''pentagrammic pyramid''' has a pentagram base and 5 intersecting triangle sides. :

Volume

The volume of a pyramid is

V= \tfrac{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 (geometry)|plane which contains the base. The formula can be formally proved using calculus: By similarity, 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

1 - \tfrac{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{B(h - y)^2}{h^2}

. The volume is given by the integral
-

\frac{B}{h^2} \int_0^h (h-y)^2 \, dy = \frac{-B}{3h^2} (h-y)^3 \bigg|_0^h = \tfrac{1}{3}Bh.

The volume of a pyramid whose base is a regular ''n''-sided polygon with side length ''s'' and whose height is ''h'' is therefore:
-

V = \frac{n}{12}hs^2 \cot\frac{\pi}{n}.

The volume of a pyramid whose base is a regular ''n''-sided polygon with radius ''R'' is therefore:
-

V = \frac{nR^2h}{6} \sin{\frac{2\pi}{n}}.

This property can be used to derive the volume for cones, as well. See also cone (geometry)|cone.

Surface area

The surface area of a pyramid is

A= B + \frac{PL}{2}

where ''B'' is the base area, ''P'' is the base perimeter and ''L'' is the slant height.

External links

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- The Uniform Polyhedra
- Angle between surfaces of a pyramid (general analytical solution), with pyramid dimensioning calculator
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- VRML models (George Hart) <3> <4> <5>
- Paper models of pyramids Category:Polyhedra Category:Self-dual polyhedra Category:Prismatoid polyhedra Category:Pyramids| Pyramid (geometry) Category:Pyramids and bipyramids| Category:Geometric shapes

Related Images

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