Size of Surface of Regular Dodecahedron

Proof

 * Euclid-XIV-3.png

Let $ABCDE$ be the regular pentagon which is the face of a regular dodecahedron.

Let the circle $ABCDE$ be circumscribed around the pentagon $ABCDE$.

Let $F$ be the center of the circle $ABCDE$.

Let $FG$ be the perpendicular dropped from $F$ to $CD$.

Let $CF$ and $FD$ be joined.

Then:

where $\map {\mathcal A} {ABCDE}$ denotes the area of the regular pentagon $ABCDE$.

The result follows from the definition of a regular dodecahedron as having $12$ such faces.