That Other Dodecahedron

Geometry Level 5

We're all familiar with the regular dodecahedron that has 12 identical flat regular pentagonal faces. If the edge length is 1, then the (area)² of each of those 12 faces is $\frac { 25+10\sqrt { 5 } }{ 16 }$ But there is another dodecahedron that also has 12 identical flat pentagonal faces. If the edge length is 1 also, what is the (area)² of each of those 12 faces? If the answer is expressed as: $\frac { a+b\sqrt { c } }{ d }$ where a, b, c, d are irreducible positive integers, find $a+b+c+d$
Those "12 identical flat pentagonal faces" need not be regular, but they must not intersect.

The answer is 28.

1 solution

Daniel Liu
May 15, 2014

The polyhedron desired is called the Concave Pyritohedral Dodecahedron . Here is the image:

Imgur Imgur

The faces are identical to the faces of a regular dodecahedron except for the fact that one pair of adjacent edges is concave, not convex. It is not hard to see that the area of a face of this shape is the area of the regular dodecahedron's face minus $ab\sin C$ where $a=b=1$ and $C=108^{\circ}$ . Evaluating gives $\sqrt{\dfrac{5+2\sqrt{5}}{16}}$ so our answer is $5+2+5+16=\boxed{28}$ .

Proving that this is the only "other" possible is much harder to do, but that probably can only be handled in a "discuss" format.

Michael Mendrin - 7 years ago

Kudos to you! For a well done answer!

Eric LeClair - 6 years, 11 months ago

I thought surface area = $15\sqrt{5 - 2\sqrt{5}}$

Sujay Sheth - 7 years ago

That Other Dodecahedron

The answer is 28.

1 solution

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