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Pentagonal Wedge


PentagonalWedgeByTruncation

The pentagonal wegde is one of the seven topologically distinct convex hexahedra. Like the cube, it contains 8 vertices, 12 edges, and 6 faces, but its faces consist of 2 triangles, 2 quadrilaterals, and 2 pentagons. As illustrated above, it can be constructed by truncating two of the corners of a tetrahedron.

PentagonalWedge

A symmetrical pentagonal wedge can be built with two regular pentagons and two equilateral triangles, leaving a single edge (the longest side of the two remaining trapezoidal faces) of different length. In particular, the long edge is a factor of the golden ratio phi times the other edge lengths.

PentagonalWedgeNet

The trapezoids of the symmetrical pentagonal wedge have angles 2pi/5 and 3pi/5, as illustrated in the net above.

This almost-unit pentagonal wedge is implemented in the Wolfram Language as PolyhedronData["PentagonalWedge"].

PentagonalWedgeCircumsphere

For short side lengths a, the symmetrical pentagonal wedge has a circumsphere of radius

 R=sin((2pi)/5)a=(2-2/(sqrt(5)))^(-1/2)a=sqrt((5+sqrt(5))/8)a
(1)

and volume

 V=1/4(2+sqrt(5))a^3.
(2)

The dihedral angle between the two pentagons is

theta=tan^(-1)2
(3)
=63.4349488... degrees.
(4)

See also

Cube, Hexahedron, Pentagonal Wedge Graph, Wedge

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References

Michon, G. P. "Final Answers: Polyhedra & Polytopes." http://nbarth.net/notes/src/notes-calc-raw/others/X-numericana/polyhedra.htm#hexahedra.

Cite this as:

Weisstein, Eric W. "Pentagonal Wedge." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PentagonalWedge.html

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