Bifrustum
In geometry, an n-gonal bifrustum is a polyhedron composed of three parallel planes of n-gons, with the middle plane largest and usually the top and bottom congruent.
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| Family of bifrusta |
|---|
| Example: hexagonal bifrustum |
| 2 n-gons2n trapezoids |
| 5n |
| 3n |
| Dnh, [n,2], (*n22) |
| n |
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| a |
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{\displaystyle {\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}}} |
| n
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{\displaystyle n{\frac {a^{2}+b^{2}+ab}{6\tan {\frac {\pi }{n}}}}h} |
| Elongated bipyramids | | convex |
In geometry, an n-gonal bifrustum is a polyhedron composed of three parallel planes of n-gons, 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.
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:
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{\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}}}
The volume V is twice the volume of a frustum.
Three bifrusta are duals to three Johnson solids, J14-16. In general, an n-gonal bifrustum has 2n trapezoids, 2 n-gons, 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 |
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