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.
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 times the other edge lengths.
The trapezoids of the symmetrical pentagonal wedge have angles and , as illustrated in the net above.
This almost-unit pentagonal wedge is implemented in the Wolfram Language as PolyhedronData["PentagonalWedge"].
For short side lengths , the symmetrical pentagonal wedge has a circumsphere of radius
(1)
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and volume
(2)
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The dihedral angle between the two pentagons is
(3)
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(4)
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