Pentagonal Wedge
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 phi times the other edge lengths.
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"].
For short side lengths a, the symmetrical pentagonal wedge has a circumsphere of radius
and volume
| V=1/4(2+sqrt(5))a^3. |
(2)
|
The dihedral angle between the two pentagons is
See also
Cube, Hexahedron, Pentagonal Wedge Graph, WedgeExplore with Wolfram|Alpha
More things to try:
References
Michon, G. P. "Final Answers: Polyhedra & Polytopes." http://nbarth.net/notes/src/notes-calc-raw/others/X-numericana/polyhedra.htm#hexahedra.Referenced on Wolfram|Alpha
Pentagonal WedgeCite this as:
Weisstein, Eric W. "Pentagonal Wedge." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PentagonalWedge.html