The radius of a circle around a regular triangle = 56 Find the height.

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

Since each angle of a regular triangle is 60 °, the area of ​​this triangle should be equal to a * a * sin (60 °) / 2 = a ^ 2 * (√3 / 2) / 2 = a ^ 2 * √3 / 4 …

According to the condition of the problem, the radius of the circle described by this triangle is 56.

Using the formula for the area of ​​a triangle in terms of the radius of the circumscribed circle, we can compose the following equation:

a ^ 3 / (4 * 56) = a ^ 2 * √3 / 4,

solving which, we get:

a ^ 3/224 = a ^ 2 * √3 / 4;

a / 224 = √3 / 4;

a = 224 * √3 / 4 = 56√3.

Applying the formula for the area of ​​a triangle in terms of the lengths of its side and the height lowered to this side, we find the height h of this regular triangle:

h = 2 * a ^ 2 * (√3 / 4) / a = 2 * a * √3 / 4 = 2 * 56√3 * √3 / 4 = 112 * 3/4 ​​= 28 * 3 = 84.

Answer: 84.