Size of Surface of Regular Icosahedron

Proof

 * Euclid-XIV-4.png

Let $ABC$ be the equilateral triangle which is the face of a regular icosahedron.

Let the circle $ABC$ be circumscribed around the equilateral triangle $ABC$.

Let $D$ be the center of the circle $ABC$.

Let $DE$ be the perpendicular dropped from $D$ to $BC$.

Let $BD$ and $CD$ be joined.

Then:

The result follows from the definition of a regular icosahedron as having $20$ such faces.