# Frustum

﻿
Frustum

: "For the graphics technique known as Frustum culling, see Hidden surface determination"A frustum [ "frustum" is Latin and means "piece, crumb". The English word is often misspelled as frustrum, probably due to the similarity with the common words "frustrate" and "frustration", also of Latin origin.] (plural: frusta or frustums) is the portion of a solid &ndash; normally a cone or pyramid &ndash; which lies between two parallel planes cutting the solid.

Elements, special cases, and related concepts

Each plane section is a "base" of the frustum. The "axis" of the frustum, 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.

Cones and pyramids can be viewed as degenerate cases of frustums, 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.

Formulas

The volume of a frustum is the difference between the volume of the cone (or other figure) before slicing the apex off, minus the volume of the cone (or other figure) that was sliced off::$V = left | frac\left\{1\right\}\left\{3\right\} h_1 B_1 - frac\left\{1\right\}\left\{3\right\} h_2 B_2 ight |.$where $h_1$ and $h_2$ are the perpendicular heights from the apex to the planes of the smaller and larger base, $B_1$, $B_2$ are the areas of the two bases.

Let $h$ be the height of the frustum, that is, the perpendicular distance between the two planes. Considering that $h = left | h_1 - h_2 ight | ,$ and $frac\left\{B_1\right\}\left\{h_1^2\right\}=frac\left\{B_2\right\}\left\{h_2^2\right\}$, one gets the alternative formula for the volume:$V = frac\left\{1\right\}\left\{3\right\} h\left(B_1+sqrt\left\{B_1 B_2\right\}+B_2\right)$(See Heronian mean.)

In particular, the volume of a circular cone frustum is :$V = frac\left\{1\right\}\left\{3\right\} pi h\left(R_1^2+R_1 R_2+R_2^2\right)$ where $pi$ is 3.14159265..., and $R_1$, $R_2$ are the radii of the two bases.

Circular Frustum

Using the definitions above, in the case of a circular frustum (or truncated cone), the volume function reduces to::$V = frac\left\{pi\right\}\left\{12\right\} h D_1^2 left\left(1 - left\left(frac\left\{D_2\right\}\left\{D_1\right\} ight\right)^2 ight\right)$ , where 'D' is the diameter of the respective base.
Equivalently::$V = frac\left\{pi\right\}\left\{12\right\} h left\left(D_1^2 - frac\left\{D_2^2\right\}\left\{D_1/D_2\right\} ight\right)$ (Although the former equation can be reduced further, this form is more intuitive.)

Also, the volume ratio can be written as a function of length ratios, or area ratios::$frac\left\{V_1\right\}\left\{V_2\right\} = left\left(frac\left\{D_1\right\}\left\{D_2\right\} ight\right)^3 = left\left(frac\left\{R_1\right\}\left\{R_2\right\} ight\right)^3 = left\left(frac\left\{h_1\right\}\left\{h_2\right\} ight\right)^3 = left\left(frac\left\{B_1\right\}\left\{B_2\right\} ight\right)^frac\left\{3\right\}\left\{2\right\}.$

Examples

* An example of a pyramidal frustum may be seen on the reverse of the Great Seal of the United States, as on the back of the U.S. one-dollar bill. The "unfinished pyramid" is surmounted by the "Eye of Providence".
* Certain ancient Native American mounds also form the frustum of a pyramid.
* 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.
* In 3D computer graphics, the usable field of view of a virtual photographic or video camera is modeled as a pyramidal frustum, the viewing frustum.

Note

*
*
* [http://www.korthalsaltes.com/special_pyramids.htm#same_height Paper models of frustums (truncated pyramids)]
* [http://www.korthalsaltes.com/tapered_cylinder.htm Paper model of frustum (truncated cone)]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Frustum — ist in der Computergrafik die mathematische Abbildung eines 3D Universums auf den Bildschirm. Frustum ist das englische Wort für ‚Kegelstumpf‘. Frustum of a pyramid bedeutet ‚Pyramidenstumpf‘. In der Fluchtpunktperspektive wird jedes Objekt in… …   Deutsch Wikipedia

• frustum — 1650s, from L. frustum piece broken off, from PIE *bhrus to , from root *bhreu to cut, break up …   Etymology dictionary

• Frustum — Frus tum, n.; pl. L. {Frusta}, E. {Frustums}. [L. fruslum piece, bit.] [1913 Webster] 1. (Geom.) The part of a solid next the base, formed by cutting off the, top; or the part of any solid, as of a cone, pyramid, etc., between two planes, which… …   The Collaborative International Dictionary of English

• frustum — [frus′təm] n. pl. frustums or frusta [frus′tə] [L, a piece, bit < IE * bhreus , to break, crush < base * bher , to split, cut > BORE] 1. a solid figure consisting of the bottom part of a cone or pyramid, the top of which has been cut off …   English World dictionary

• Frustum — Se ha sugerido que este artículo o sección sea fusionado con Tronco (geometría) (discusión). Una vez que hayas realizado la fusión de artículos, pide la fusión de historiales aquí. Para el sólido geométrico, véase Tronco (geometría) …   Wikipedia Español

• Frustum — Frus|tum das; s, ...ta <aus gleichbed. lat. frustum> (veraltet) Stück, Brocken, Bissen …   Das große Fremdwörterbuch

• frustum — noun (plural frustums or frusta) Etymology: New Latin, from Latin, piece, bit more at bruise Date: 1658 the basal part of a solid cone or pyramid formed by cutting off the top by a plane parallel to the base; also the part of a solid intersected… …   New Collegiate Dictionary

• frustum — /frus teuhm/, n., pl. frustums, frusta / teuh/. Geom. 1. the part of a conical solid left after cutting off a top portion with a plane parallel to the base. 2. the part of a solid, as a cone or pyramid, between two usually parallel cutting planes …   Universalium

• frustum — noun a) A cone or pyramid whose tip has been truncated by a plane parallel to its base. b) A region of a sphere delimited by two parallel planes. Syn …   Wiktionary

• frustum — frusÂ·tum || frÊŒstÉ™m n. (Geometry) truncated cone or pyramid; part of a solid that is cut parallel …   English contemporary dictionary