The difference between the lengths of the circumcircle and the incircle

The difference between the lengths of the circumcircle and the incircle of a regular triangle is 2√3п cm. Find the area of the triangle.

We denote by a the length of the side of this regular triangle.

Then the area of ​​this triangle should be equal to a ^ 2 * sin (60 °) / 2 = a ^ 2 * √3 / 4 cm ^ 2.

Applying the formulas for the area of ​​a triangle through the radii of the inscribed and circumscribed circles, we express the radii of the inscribed and circumscribed circles of this triangle through a:

r = a ^ 2 * √3 / 4 / (3a / 2) = a ^ 2 * √3 / 4 * 2 / 3a = a√3 / 6;

R = a ^ 3 / (4 * a ^ 2 * √3 / 4) = a ^ 3 / (a ​​^ 2 * √3) = a / √3 = a√3 / 3.

According to the condition of the problem, the difference between the lengths of the circumscribed and inscribed circles of this triangle is 2√3, therefore, we can compose the following equation:

a√3 / 3 – a√3 / 6 = 2√3,

solving which, we get:

6 * (a√3 / 3 – a√3 / 6) = 6 * 2√3;

2а√3 – а√3 = 12√3;

a√3 = 12√3;

a = 12√3 / √3 = 12 cm.

Find the area of ​​the triangle:

a ^ 2 * √3 / 4 = 12 ^ 2 * √3 / 4 = 144 * √3 / 4 = 36√3 cm ^ 2.

Answer: 36√3 cm ^ 2.