Bifrustum

{{short description|Polyhedron made by joining two identical frusta at their bases}} {{Infobox polyhedron | name = Family of bifrusta | image = Hexagonal bifrustum.png | caption = Example: hexagonal bifrustum | type = <!--general type of this shape--> | euler = <!--Euler characteristic--> | faces = {{math|2}} {{mvar|n}}-gons<br>{{math|2''n''}} trapezoids | edges = {{math|5''n''}} | vertices = {{math|3''n''}} | vertex_config = <!--list faces around a vertex--> | schläfli = <!--Schläfli symbol--> | wythoff = <!--Wythoff symbol--> | conway = <!--Conway polyhedron notation--> | coxeter = <!--Coxeter-Dynkin diagram--> | symmetry = {{math|''D''_''nh'', [''n'',2], (*''n''22)}} | rotation_group = <!--rotation group--> | surface_area = <math>\begin{align} &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{align}</math> | volume = <math>n \frac{a^2+b^2+ab}{6 \tan{\frac{\pi}{n}}}h</math> | angle = <!--dihedral angle--> | dual = Elongated bipyramids | properties = convex | vertex_figure = <!--only image file name--> | net = <!--only image file name--> | net_caption = <!--net caption--> }}

In geometry, an {{mvar|n}}-gonal '''bifrustum''' is a polyhedron composed of three parallel planes of {{mvar|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.<ref>{{Cite web |title=Octagonal Bifrustum |url=https://etc.usf.edu/clipart/42700/42718/bifrustum-02_42718.htm |access-date=2022-06-16 |website=etc.usf.edu |language=en}}</ref>

They are duals to the family of elongated bipyramids.

== Formulae == For a regular {{mvar|n}}-gonal bifrustum with the equatorial polygon sides {{mvar|a}}, bases sides {{mvar|b}} and semi-height (half the distance between the planes of bases) {{mvar|h}}, the lateral surface area {{mvar|A_l}}, total area {{mvar|A}} and volume {{mvar|V}} are:<ref name=rechner>{{cite web |url=https://rechneronline.de/pi/bifrustum.php |title=Regelmäßiges Bifrustum - Rechner |website=RECHNERonline |language=de |access-date=2022-06-30}}</ref> <math display=block>\begin{align} 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{align}</math> The volume V is twice the volume of a frustum.

== Forms == Three bifrusta are duals to three Johnson solids, {{math|''J''{{sub|14-16}}}}. In general, an '''{{mvar|n}}-gonal bifrustum''' has {{math|2''n''}} trapezoids, 2 {{mvar|n}}-gons, and is dual to the elongated dipyramids.

{| class=wikitable width=450 !Triangular bifrustum !Square bifrustum !Pentagonal bifrustum |- align=center |150px |150px |150px |- valign=top |6 trapezoids, 2 triangles. Dual to elongated triangular bipyramid, {{math|''J''{{sub|14}}}} |8 trapezoids, 2 squares. Dual to elongated square bipyramid, {{math|''J''{{sub|15}}}} |10 trapezoids, 2 pentagons. Dual to elongated pentagonal bipyramid, {{math|''J''{{sub|16}}}} |}

== References == <references />

== External links == * {{MathWorld |title=Pyramidal Frustum |id=PyramidalFrustum |access-date=1 December 2023}}

{{Polyhedron navigator}} Category:Polyhedra {{Polyhedron-stub}}