The ratio of the areas of the incircle and circumcircle of an equilateral triangle is

### Concept & Area Ratio of Circles The area of a circle is proportional to the square of its radius ($A = \pi r^2$). Therefore, the ratio of the areas of any two circles is equal to the square of the ratio of their radii. $$ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 $$ ### Step-by-Step Solution * Let the side of the equilateral triangle be $a$. * The radius of the incircle is $r = \frac{a}{2\sqrt{3}}$. * The radius of the circumcircle is $R = \frac{a}{\sqrt{3}}$. * The ratio of their radii is $r : R = \frac{a}{2\sqrt{3}} : \frac{a}{\sqrt{3}} = 1 : 2$. * The area of the incircle is $\pi r^2$. * The area of the circumcircle is $\pi R^2$. * The ratio of their areas is $\frac{\pi r^2}{\pi R^2} = \left(\frac{r}{R}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$. * Therefore, the ratio is $1 : 4$. ### Exam Strategy & Shortcut Knowing the fixed geometric property that the circumradius is twice the inradius ($R = 2r$) for an equilateral triangle, you can directly square this $1:2$ linear ratio to get the $1:4$ area ratio. ### Common Pitfall Students often confuse the linear ratio of the radii ($1:2$) with the ratio of the areas. Always remember to square the 1D ratio to find the 2D area ratio. ### Final Answer Therefore, the correct answer is **$1 : 4$**.