The radius of a circle inscribed in an equilateral triangle is 12. Find the height of this triangle.

Let x denote the side length of this equilateral triangle.

In the initial data for this task, it is reported that the radius of the circle inscribed in this triangle is 12.

Let us express the area S of this triangle through the radius of the inscribed circle:

S = 12 * (x + x + x) / 2 = 12 * 3x / 2 = 6 * 3x = 18x.

Let us express the area S of this triangle through two sides about the angle between them:

S = (1/2) * x * x * sin (60 °) = (1/2) * x ^ 2 * √3 / 2 = x ^ 2√3 / 4.

We compose and solve the equation:

x ^ 2√3 / 4 = 18x;

x ^ 2 / x = 18 * 4 / √3;

x = 72 / √3.

Using the formula for the area of ​​a triangle in terms of height and base, we find the height of this triangle:

h = 2 * S / x = 2 * 18x / x = 2 * 18 = 36.

Answer: 36.