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 Al, total area A and volume V are:2 and 3 A l = n ( a + b ) ( a − b 2 cot π n ) 2 + h 2 A = A l + n b 2 2 tan π n V = n a 2 + b 2 + a b 6 tan π n h {\displaystyle {\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, J14-16. 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. https://etc.usf.edu/clipart/42700/42718/bifrustum-02_42718.htm ↩
"Regelmäßiges Bifrustum - Rechner". RECHNERonline (in German). Retrieved 2022-06-30. https://rechneronline.de/pi/bifrustum.php ↩
"mathworld pyramidal frustum". https://mathworld.wolfram.com/PyramidalFrustum.html ↩