An equilateral triangle with a side of 6 cm is inscribed in the cross section of the ball.

An equilateral triangle with a side of 6 cm is inscribed in the cross section of the ball. The distance from the center of the ball to the plane of the triangle is 2 cm. Find the radius of the ball?

Let’s denote the inscribed equilateral triangle ABC, the distance from the center of the ball to the plane OO1 = 2 cm.

The sides of the triangle are equal to a = 6 cm.

Point O is the center of the smaller circle in which the triangle ABC is inscribed, point O1 is the center of the ball and the larger circle.

Since an equilateral triangle is inscribed in a circle, we find the radius of the smaller ball by the formula

r = a / √3 = 6 / √3 = 2√3 cm.

Consider a right-angled triangle AO1O, where AO1 is the radius of the ball R, AO = r = 2√3 cm.

Then, by the Pythagorean theorem, the radius of the ball is:

AO1 = R = ((OO1) ^ 2 + (AO) ^ 2) ^ (1/2) = ((2) ^ 2 + (2√3) ^ 2) ^ (1/2) = (4 + 4 * 3) ^ (1/2) = √16 = 4 cm.