Frustum

Set of pyramidal frusta

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

In geometry, a frustum (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.

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.

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

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

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.

External links

• Derivation of formula for the volume of frustums of pyramid and cone (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)