# Pyramid (geometry)

﻿
Pyramid (geometry)

:"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" &ge; 3). In other words, it is a conic solid with polygonal base.

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 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. All pyramids are self-dual.

Pyramids are a subclass of the prismatoids. The 1-skeleton of pyramid is a wheel graph.

Volume

The volume of a pyramid is $V = frac\left\{1\right\}\left\{3\right\} Bh$ 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 $frac\left\{h-y\right\}\left\{h\right\}$, 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 $frac\left\{A\right\}\left\{h^2\right\}\left(h-y\right)^2$.:The volume is given by the integral

(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.)

Surface area

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

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:

ee also

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

*
*
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.slyman.org/right_projects_math.php Angle between surfaces of a pyramid (general analytical solution), Pyramid dimensioning calculator] at [http://www.slyman.org/ www.slyman.org]
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
** VRML models [http://www.georgehart.com/virtual-polyhedra/alphabetic-list.html (George Hart)] [http://www.georgehart.com/virtual-polyhedra/vrml/tetrahedron.wrl <3>] [http://www.georgehart.com/virtual-polyhedra/vrml/square_pyramid_(J1).wrl <4>] [http://www.georgehart.com/virtual-polyhedra/vrml/pentagonal_pyramid_(J2).wrl <5>]
* [http://www.korthalsaltes.com/special_pyramids.htm Paper models of pyramids]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Pyramid — This article is about pyramid shaped structures. For the geometric term, see Pyramid (geometry). For other uses, see Pyramid (disambiguation). The Egyptian pyramids of the Giza Necropolis, as seen from above …   Wikipedia

• Pyramid (disambiguation) — A pyramid is any three dimensional structure where the upper surfaces are triangular and converge at one point. Pyramid may also refer to:* Pyramid (geometry), a conic solid with polygonal base * Egyptian pyramids, the pyramids of Egypt **… …   Wikipedia

• Geometry — (Greek γεωμετρία ; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of… …   Wikipedia

• pyramid — ► NOUN 1) a monumental stone structure with a square or triangular base and sloping sides that meet in a point at the top, especially one built as a royal tomb in ancient Egypt. 2) Geometry a polyhedron of which one face is a polygon and the… …   English terms dictionary

• geometry — /jee om i tree/, n. 1. the branch of mathematics that deals with the deduction of the properties, measurement, and relationships of points, lines, angles, and figures in space from their defining conditions by means of certain assumed properties… …   Universalium

• Pyramid inch — The pyramid inch, infrequently called the sacred Jewish inchFact|date=July 2007, is a unit of measure claimed by pyramidologists to have been used in ancient times. Supposedly it was one twenty fifth of a sacred cubit , 1.00106 British inches, or …   Wikipedia

• pyramid — /ˈpɪrəmɪd / (say piruhmid) noun 1. Architecture a massive structure built of stone, with square (or polygonal) base, and sloping sides meeting at an apex, such as those built by the ancient Egyptians as royal tombs or by the Mayas as platforms… …   Australian English dictionary

• pyramid — [ pɪrəmɪd] noun 1》 a monumental stone structure with a square or triangular base and sloping sides that meet in a point at the top, especially one built as a royal tomb in ancient Egypt. 2》 Geometry a polyhedron of which one face is a polygon and …   English new terms dictionary

• Cone (geometry) — Not to be confused with the conical surface.. A right circular cone and an oblique circular cone A cone is an n dimensional geometric shape that tapers smoothly from a base (usually flat and circular) to a point called the apex or vertex.… …   Wikipedia

• History of geometry — Geometry (Greek γεωμετρία ; geo = earth, metria = measure) 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. Classic geometry… …   Wikipedia