Bifrustum

Family of bifrusta
Example: hexagonal bifrustum
Faces	2 n-gons 2n trapezoids
Edges	5n
Vertices	3n
Symmetry group	Dnh, [n,2], (*n22)
Surface area	$\begin{array}{cc}& n(a+b)\sqrt{{(\frac{a-b}{2}\cot \frac{\pi }{n})}^2+h^2}\\ & +n\frac{b^2}{2\tan \frac{\pi }{n}}\end{array}$
Volume	$n\frac{a^2+b^2+ab}{6\tan \frac{\pi }{n}}h$
Dual polyhedron	Elongated bipyramids
Properties	convex

In geometry, an n-agonal bifrustum is a polyhedron composed of three parallel planes of n-agons, with the middle plane largest and usually the top and bottom congruent.

It can be constructed as two congruent frusta combined across a plane of symmetry, and also as a bipyramid with the two polar vertices truncated.

They are duals to the family of elongated bipyramids.

Formulae

For a regular n-gonal bifrustum with the equatorial polygon sides a, bases sides b and semi-height (half the distance between the planes of bases) h, the lateral surface area A_l, total area A and volume V are: and $$\begin{array}{cc}{A}_l& =n(a+b)\sqrt{{(\frac{a-b}{2}\cot \frac{\pi }{n})}^2+h^2}\\ A& =A_l+n\frac{b^2}{2\tan \frac{\pi }{n}}\\ V& =n\frac{a^2+b^2+ab}{6\tan \frac{\pi }{n}}h\end{array}$$ Note that the volume V is twice the volume of a frusta.

Forms

Three bifrusta are duals to three Johnson solids, J_14-16. In general, a n-agonal bifrustum has 2n trapezoids, 2 n-agons, and is dual to the elongated dipyramids.

Triangular bifrustum	Square bifrustum	Pentagonal bifrustum
6 trapezoids, 2 triangles. Dual to elongated triangular bipyramid, J14	8 trapezoids, 2 squares. Dual to elongated square bipyramid, J15	10 trapezoids, 2 pentagons. Dual to elongated pentagonal bipyramid, J16