# Find the ratio of the areas of the in-circle and circum-circle of an equilateral triangle?

Nov 5, 2016

$1 : 4$

#### Explanation:

Given $\Delta A B C =$ equilateral triangle
Let radius of in-circle be $r$, and radius of circumcircle be $R$.
In $\Delta O B D , \angle O B D = {30}^{\circ} , \angle O D B = {90}^{\circ} \implies R = 2 r$

Let area of in-circle be ${A}_{I}$ and area of circumcircle be ${A}_{C}$,

$\implies {A}_{I} / {A}_{C} = \frac{\pi {r}^{2}}{\pi \left({R}^{2}\right)} = \frac{\pi {r}^{2}}{\pi {\left(2 r\right)}^{2}} = \frac{1}{4}$

Hence, ${A}_{I} : {A}_{C} = 1 : 4$