Question

The ratio of the area of the circumcircle and incircle of an equilateral triangle is:

• A. $4:1$
• B. $1:4$
• C. $9:1$
• D. $1:9$

Answer 1

The correct option is A $4:1$

For an equilateral triangle, $s=\frac{a+b+c}{2}=\frac{3a}{2}$ and $\Delta =\frac{\sqrt{3}}{4}{a}^2$

$r^2={\left(\frac{\Delta }{s}\right)}^2=\frac{1}{12}{a}^2$

$R^2={\left(\frac{abc}{4\Delta }\right)}^2=\frac{1}{3}{a}^2$

$\therefore \frac{\text{Area of Circumcircle}}{\text{Area of Incircle}}=\frac{\pi R^2}{\pi r^2}=\frac{a^2}{3}\times \frac{12}{a^2}=4$