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Bifrustum

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Polyhedron made by joining two identical frusta at their bases

Family of bifrusta
Example: hexagonal bifrustum
Faces	2n-gons 2ntrapezoids
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

Ingeometry, ann-agonalbifrustum is apolyhedron composed of three parallel planes ofn-agons, with the middle plane largest and usually the top and bottom congruent.

It can be constructed as two congruentfrusta combined across a plane of symmetry, and also as abipyramid with the two polar vertices truncated.^[1]

They areduals to the family ofelongated bipyramids.

Formulae

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For a regularn-gonal bifrustum with the equatorial polygon sidesa, bases sidesb and semi-height (half the distance between the planes of bases)h, thelateral surface areaA_l, total areaA andvolumeV 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 afrusta.

Forms

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Three bifrusta areduals to threeJohnson solids,J_14-16. In general, an-agonal bifrustum has2n trapezoids, 2n-agons, and is dual to theelongated dipyramids.

Triangular bifrustum	Square bifrustum	Pentagonal bifrustum

6 trapezoids, 2 triangles. Dual toelongated triangular bipyramid,J₁₄	8 trapezoids, 2 squares. Dual toelongated square bipyramid,J₁₅	10 trapezoids, 2 pentagons. Dual toelongated pentagonal bipyramid,J₁₆