Question

A point is taken at random from inside of the circumcircle of an equilateral triangle. The probability that it lies inside the circumcircle but outside the incircle is?

• A. $\frac{1}{4}$
• B. $\frac{3}{4}$
• C. $\frac{1}{2}$
• D. $\frac{1}{3}$
  

Answer 1

The correct option is B $\frac{3}{4}$

Consider an equilateral triangle of side $2a$.

Radius of circumcircle $R=a\left(\sec {30}^o\right)=\frac{2a}{\sqrt{3}}$

Radius of incircle $r=a\left(\tan {30}^o\right)=\frac{a}{\sqrt{3}}$

The probability that point lies inside the circumcircle but outside the incircle is $=$ (area inside circumcircle but outside incircle)$/$ (area of circumcircle)

$=\frac{\pi R^2-\pi r^2}{\pi R^2}$

$=\frac{\frac{4a^2}{3}-\frac{a^2}{3}}{\frac{4a^2}{3}}$

$=\frac{3}{4}$

Hence $B$ is correct option