frustum

Set of pyramidal frusta
Examples: Pentagonal and square frusta
Faces	n trapezoids, 2 n-gons
Edges	3n
Vertices	2n
Symmetry group	Cnv
Dual polyhedron	-
Properties	convex

For the graphics technique known as Frustum culling, see Hidden surface determination

In geometry, a frustum ^[1] (plural: frusta or frustums) is the portion of a solid — normally a cone or pyramid — which 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 (which is formed by a clipped pyramid). In the aerospace industry, it is used for the fairing between two stages of a multistage rocket (such as the Saturn V), which is shaped like a truncated cone.

Elements, special cases, and related concepts

Each plane section is a floor 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.

Formulae

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:

where h₁ and h₂ are the perpendicular heights from the apex to the planes of the smaller and larger base, B₁, B₂ 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 and , one gets the alternative formula for the volume

(See Heronian mean.)

For a regular polygon:

In particular, the volume of a circular cone frustum is

where π is 3.14159265..., and R₁, R₂ are the radii 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 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.
• 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.

Note

1. ^ frustum is Latin and means piece, 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.

External links

Look up frustum in Wiktionary, the free dictionary.

Wikimedia Commons has media related to: Frustums

• Derivation of formula for the volume of frustums of pyramid and cone at Mathalino.com
• Weisstein, Eric W., "Pyramidal frustum" from MathWorld.
• Weisstein, Eric W., "Conical frustum" from MathWorld.
• Paper models of frustums (truncated pyramids)
• Paper model of frustum (truncated cone)
• Design paper models of conical frustum (truncated cones)