Frustum

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Set of pyramidal frusta
Image:Pentagonal frustum.png
Faces	n trapezoids, 2 n-gon
Edges	3n
Vertices	2n
Symmetry group	C_nv
Dual polyhedron	-
Properties	convex

A frustum is the portion of a solid – normally a cone or pyramid – which lies between two parallel planes cutting the solid. Degenerate cases are obtained for finite solids by cutting with a single plane only.

Pyramidal frusta are a subclass of the prismatoids.

The formula for the volume of the frustum is

<math>V =\frac{1}{3} h(B_1+\sqrt{B_1\times B_2}+B_2)</math>

where h is the height from the top base to the bottom base, B₁ is the area of the bottom base, and B₂ is the area of the top base. (See also: Heronian mean.) A more intuitive formula is: the volume of the cone (or other figure) before you sliced the top off, minus the volume of the cone (or other figure) that you sliced off:

The result comes from <math>h = h_1 - h_2\,</math> and <math>\frac{B_1}{h_1^2}=\frac{B_2}{h_2^2}</math>.

Two frusta joined at their bases make a bifrustum.

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 focal field of a still or video camera forms a frustum. In 3D computer graphics, this is called the viewing frustum.

The spelling frustrum, listed as "erroneous" by the Oxford English Dictionary, is frequently encountered and might be considered a variant. The Oxford English Dictionary gives both frusta and frustums for the plural.