The area of a regular triangle is greater than the area of the inscribed circle by 15 square roots of 3.

The area of a regular triangle is greater than the area of the inscribed circle by 15 square roots of 3. Find the radius of the circle.

As we know, the area of the specified figure can be calculated through:

S = √3 / 4 * a ^ 2.

At the same time, from the school geometry course, we remember that the radius of a circle inscribed in such a figure can be calculated using:

r = √3 / 6 * a.

Therefore, the area of the circle will be equal to:

S = π (√3 / 6 * a) ^ 2 = π * 3/36 * a ^ 2 = π / 1 ^ 2 * a.

Since it is known that the first area is 15√3 larger than the second, then we write the equation and find the square of the side:

√3 / 4 * a ^ 2 – π / 1 ^ 2 * a ^ 2 = 15√3;

a2 = 15√3 / (√3 / 4 – π / 1 ^ 2).

Then the radius is equal to:

√3 / 6 * √ (15√3 / (√3 / 4 – π / 1 ^ 2)).

Answer: √3 / 6 * √ (15√3 / (√3 / 4 – π / 1 ^ 2)).