Bifrustum

Polyhedron made by joining two identical frusta at their bases

Family of bifrusta
Example: hexagonal bifrustum
Faces	2 n-gons 2n trapezoids
Edges	5n
Vertices	3n
Symmetry group	D_nh, [n,2], (*n22)
Surface area	${\begin{aligned}&n(a+b){\sqrt {\left({\tfrac {a-b}{2}}\cot {\tfrac {\pi }{n}}\right)^{2}+h^{2}}}\\[2pt]&\ \ +\ n{\frac {b^{2}}{2\tan {\frac {\pi }{n}}}}\end{aligned}}$
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.^[1]

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:^[2] and ^[3]

{\begin{aligned}A_{l}&=n(a+b){\sqrt {\left({\tfrac {a-b}{2}}\cot {\tfrac {\pi }{n}}\right)^{2}+h^{2}}}\\[4pt]A&=A_{l}+n{\frac {b^{2}}{2\tan {\frac {\pi }{n}}}}\\[4pt]V&=n{\frac {a^{2}+b^{2}+ab}{6\tan {\frac {\pi }{n}}}}h\end{aligned}}

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

References

^ "Octagonal Bifrustum". etc.usf.edu. Retrieved 2022-06-16.
^ "Regelmäßiges Bifrustum - Rechner". RECHNERonline (in German). Retrieved 2022-06-30.
^ "mathworld pyramidal frustum".