The radius of a circle circumscribed about a regular triangle is 56. Find the height of this triangle

The radius of a circle circumscribed about a regular triangle is determined by the formula:

R = a / √3.

Find the side of this triangle:

a = R * √3 = 56√3.

In a regular triangle, all heights are equal to each other. In addition, the height is both the median and divides the side to which it is drawn in half. Thus, the height, side and half of the side form a right-angled triangle, for which, according to the Pythagorean theorem, the statement is true:

h ^ 2 + (a / 2) ^ 2 = a ^ 2.

Hence:

h ^ 2 = a ^ 2 – (a / 2) ^ 2 = (56√3) ^ 2 – (56√3 / 2) ^ 2 = (56√3) ^ 2 – (28√3) ^ 2 = 3136 * 3 – 784 * 3 = 2352 * 3;

h = √ (2352 * 3) = √ (784 * 3 * 3) = 28 * 3 = 84.