Pentagonal Wedge

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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 [].

👁 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