frustum

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frustum

frustum of a pyramid
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(frŭs'təm) pronunciation

n. Mathematics, pl., -tums, or -ta (-tə).
The part of a solid, such as a cone or pyramid, between two parallel planes cutting the solid, especially the section between the base and a plane parallel to the base.

[Latin frūstum, piece broken off.]

frustum

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1. Slice of a solid body, especially a form produced by cutting through a cone or pyramid between the base and a parallel plane, or between any two planes.

2. Drum of a column-shaft.

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For a list of words related to frustum, see:

Geometric Shapes and Mathematically Defined Forms - frustum: conical solid with top cut off by plane parallel to base
Shapes - frustum: part of solid cone or pyramid left by cutting off top portion by plane parallel to base

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Frustum

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Set of pyramidal frusta
Examples: Pentagonal and square frusta
Faces	n trapezoids, 2 n-gons
Edges	3n
Vertices	2n
Symmetry group	C_nv, [1,n], (*nn)
Properties	convex

In geometry, a frustum^[1] (plural: frusta or frustums) is the portion of a solid (normally a cone or pyramid) that lies between two parallel planes cutting it.

The term is commonly used in computer graphics to describe the three-dimensional region which is visible on the screen, the "viewing frustum", which is formed by a clipped pyramid; in particular, frustum culling is a method of hidden surface determination.

In the aerospace industry, frustum is the common term for the fairing between two stages of a multistage rocket (such as the Saturn V), which is shaped like a truncated cone.

Contents

1 Elements, special cases, and related concepts
2 Formula
- 2.1 Volume
- 2.2 Surface area
3 Examples
4 Notes
5 External links

Elements, special cases, and related concepts

Each plane section is a floor or base of the frustum. Its axis if any, is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.

The height of a frustum is the perpendicular distance between the planes of the two bases.

Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the apex (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of the prismatoids.

Two frusta joined at their bases make a bifrustum.

Formula

Volume

The volume of a conical or pyramidal frustum is the volume of the solid before slicing the apex off, minus the volume of the apex:

$V = \frac{h_2 B_2 - h_1 B_1}{3}$

where B₁ is the area of one base, B₂ is the area of the other base, and h₁, h₂ are the perpendicular heights from the apex to the planes of the two bases.

Considering that

$\frac{B_1}{h_1^2}=\frac{B_2}{h_2^2}$

the volume can also be expressed as the product of the height h = h₂−h₁ of the frustum, and the Heronian mean of their areas:

$V = \frac{h}{3}(B_1+B_2+\sqrt{B_1 B_2})$

Heron of Alexandria is noted for deriving this formula and with it encountering the imaginary number, the square root of negative one.^[2]

In particular, the volume of a circular cone frustum is

$V = \frac{\pi h}{3}(R_1^2+R_2^2+R_1 R_2)$

where π is 3.14159265..., and R₁, R₂ are the radii of the two bases.

The volume of a pyramidal frustum whose bases are n-sided regular polygons is

$V= \frac{n h}{12} (a_1^2+a_2^2+a_1a_2)\cot \frac{180}{n}$

where a₁ and a₂ are the sides of the two bases.

Surface area

For a right circular conical frustum^[3]

$\begin{align}\text{Lateral Surface Area}&=\pi(R_1+R_2)s\\ &=\pi(R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}\end{align}$

and

$\begin{align}\text{Total Surface Area}&=\pi((R_1+R_2)s+R_1^2+R_2^2)\\ &=\pi((R_1+R_2)\sqrt{(R_1-R_2)^2+h^2}+R_1^2+R_2^2)\end{align}$

where R₁ and R₂ are the base and top radii respectively, and s is the slant height of the frustum.

The surface area of a right frustum whose bases are similar regular n-sided polygons is

$A= \frac{n}{4}\left[(a_1^2+a_2^2)\cot \frac{\pi}{n} + \sqrt{(a_1^2-a_2^2)^2\sec^2 \frac{\pi}{n}+4 h^2(a_1+a_2)^2} \right]$

where a₁ and a₂ are the sides of the two bases.

Examples

A notable example of a pyramidal frustum appears on the reverse of the Great Seal of the United States on the back of the United States one-dollar bill, the "unfinished pyramid" being surmounted by the Eye of Providence.
Certain ancient Native American mounds also form the frustum of a pyramid.
Chinese pyramids
The John Hancock Center in Chicago, Illinois is a frustum whose bases are rectangles.
The Washington Monument is a narrow pyramidal frustum (with square bases) with a pyramid attached to the top base.
The viewing frustum in 3D computer graphics is the usable field of view of a virtual photographic or video camera modeled as a pyramidal frustum.
The poem Love and tensor algebra, in the English translation of Stanislaw Lem's short-story collection The Cyberiad, claims that every frustum longs to be a cone.
A bucket is an everyday example of a conical frustum. The bottom internal diameter is usually smaller than the upper internal diameter.
The stereotypical lampshade is a frustum.

Notes

^ The term comes from Latin frustum meaning "piece" or "crumb". The English word is often misspelled as frustrum, probably because of a similarity with the common words "frustrate" and "frustration", also of Latin origin, or "fulcrum"
^ Nahin, Paul. "An Imaginary Tale: The story of [the square root of minus one]." Princeton University Press. 1998
^ "Mathwords.com: Frustum". http://www.mathwords.com/f/frustum.htm. Retrieved 17 July 2011.

External links

Look up frustum in Wiktionary, the free dictionary.

Wikimedia Commons has media related to: Frustums

v t e Polyhedron navigator

Platonic solids (regular)	tetrahedron · cube · octahedron · dodecahedron · icosahedron

Archimedean solids (Semiregular/Uniform)	truncated tetrahedron · cuboctahedron · truncated cube · truncated octahedron · rhombicuboctahedron · truncated cuboctahedron · snub cube · icosidodecahedron · truncated dodecahedron · truncated icosahedron · rhombicosidodecahedron · truncated icosidodecahedron · snub dodecahedron

Catalan solids (Dual semiregular)	triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedron

Dihedral regular	dihedron · hosohedron

Dihedral uniform	prisms · antiprisms

Duals of dihedral uniform	bipyramids · trapezohedra

Dihedral others	pyramids · truncated trapezohedra · gyroelongated bipyramid · cupola · bicupola · pyramidal frusta

Degenerate polyhedra are in italics.

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American Heritage Dictionary The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read

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