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 – normally a cone or pyramid – 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::$V\; =\; left\; |\; frac\{1\}\{3\}\; h\_1\; B\_1\; -\; frac\{1\}\{3\}\; h\_2\; B\_2\; ight\; |.$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 $frac\{B\_1\}\{h\_1^2\}=frac\{B\_2\}\{h\_2^2\}$, one gets the alternative formula for the volume:$V\; =\; frac\{1\}\{3\}\; h(B\_1+sqrt\{B\_1\; B\_2\}+B\_2)$(See Heronian mean.)

In particular, the volume of a circular cone frustum is :$V\; =\; frac\{1\}\{3\}\; pi\; h(R\_1^2+R\_1\; R\_2+R\_2^2)$ 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::$V\; =\; frac\{pi\}\{12\}\; h\; D\_1^2\; left(1\; -\; left(frac\{D\_2\}\{D\_1\}\; ight)^2\; ight)$ , where 'D' is the diameter of the respective base.
Equivalently::$V\; =\; frac\{pi\}\{12\}\; h\; left(D\_1^2\; -\; frac\{D\_2^2\}\{D\_1/D\_2\}\; ight)$ (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::$frac\{V\_1\}\{V\_2\}\; =\; left(frac\{D\_1\}\{D\_2\}\; ight)^3\; =\; left(frac\{R\_1\}\{R\_2\}\; ight)^3\; =\; left(frac\{h\_1\}\{h\_2\}\; ight)^3\; =\; left(frac\{B\_1\}\{B\_2\}\; ight)^frac\{3\}\{2\}.$

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

External links

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* [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)]