Question

There is an equilateral triangle with each side 4 unit and a circle with the center on one of the vertex of that triangle. The arc of that circle divides the triangle into two parts of equal area. How long is the radius of the circle?

Answer 1

Area of equilateral triangle $=\frac{\sqrt{3}}{4}(4{)}^2=4\sqrt{3}$
Let radius of the circle be $a$.
Thus using given condition, $2\sqrt{3}=\pi a^2\times \frac{60}{360}=\frac{\pi }{6}.a^2\Rightarrow a=\sqrt{\frac{12\sqrt{3}}{\pi }}$