Size of Surface of Regular Icosahedron
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Theorem
In the words of Hypsicles of Alexandria:
- If $ABC$ be an equilateral triangle in a circle, $D$ the centre, and $DE$ perpendicular to $BC$,
$30 BC . DE =$ (surface of icosahedron).
(The Elements: Book $\text{XIV}$: Proposition $4$)
Proof
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:
\(\ds DE \cdot BC\) | \(=\) | \(\ds 2 \cdot \triangle DBC\) | Area of Triangle in Terms of Side and Altitude | |||||||||||
\(\ds \therefore \ \ \) | \(\ds 3 \cdot DE \cdot BC\) | \(=\) | \(\ds 6 \cdot \triangle DBC\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \cdot \triangle ABC\) | ||||||||||||
\(\ds \therefore \ \ \) | \(\ds 30 \cdot DE \cdot BC\) | \(=\) | \(\ds 20 \cdot \triangle ABC\) |
The result follows from the definition of a regular icosahedron as having $20$ such faces.
$\blacksquare$
Historical Note
This proof is Proposition $4$ of Book $\text{XIV}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): The So-Called Book $\text{XIV}$, by Hypsicles