Find the area of an equilateral triangle if the radius of the circle inscribed in it is √3.

Let us denote by a the length of the sides of this equilateral triangle.

Since each angle of any equilateral triangle is 60 °, the area S of this triangle should be equal to:

S = a * a * sin (60 °) / 2 = a ^ 2 * (√3 / 2) / 2 = a ^ 2 * √3 / 4.

Also in the condition of the problem it is said that the radius of the circle inscribed in the given triangle is equal to √3, therefore, we can compose the following equation:

(3a / 2) * √3 = a ^ 2 * √3 / 4;

solving which, we get:

(3/2) * √3 = a * √3 / 4;

a = (3/2) * √3 / (√3 / 4) = (3/2) * √3 * 4 / √3 = 6.

Therefore, the area of ​​this triangle is a ^ 2 * √3 / 4 = 6 ^ 2 * √3 / 4 = 36 * √3 / 4 = 9√3.

Answer: 9√3.