Equilateral triangle

In the equilateral triangle the height, the radius of the circumscribed circle and the radius of the inscribed circle are related by simple, memorable ratios.

Property — Equilateral triangle of side $\ell$

$h = \frac{\ell\sqrt{3}}{2}, \qquad R = \frac{2}{3}\,h = \frac{\ell\sqrt{3}}{3}, \qquad r = \frac{1}{3}\,h = \frac{\ell\sqrt{3}}{6}.$ where $R$ is the radius of the circumscribed circle and $r$ that of the inscribed circle.

Example — Equilateral of side 6

In an equilateral triangle with $\ell = 6$: $h = 3\sqrt{3}, \qquad R = 2\sqrt{3}, \qquad r = \sqrt{3}.$ Area: $S = \dfrac{\ell^2\sqrt{3}}{4} = \dfrac{36\sqrt{3}}{4} = \boxed{9\sqrt{3}}$.

Check with the formula $r=S/p$: $r = \dfrac{9\sqrt{3}}{9} = \sqrt{3}$ ✓

Links

Topics: Euclidean circle
Concepts: Circumscribed circle · Inscribed circle · Equilateral triangle
Skills: Calculating · Using formulae