A regular triangular prism is inscribed in the cylinder, the diagonal of the side face of which is equal to the root of 507

A regular triangular prism is inscribed in the cylinder, the diagonal of the side face of which is equal to the root of 507. The radius of the base of the cylinder is 11. Find the height of the prism.

1. In an equilateral triangle ABC, draw medians AM and BN, which are also heights:

AN = 1/2 * AC = a / 2.
2. The medians of a regular triangle intersect at the center of the circumcircle and are divided in a ratio of 2: 1:

BO: NO = 2: 1, hence:
NO = 1/2 * BO = R / 2;
BN = BO + NO = R + R / 2 = 3R / 2.
3. Apply the Pythagorean theorem to the right-angled triangle ABN:

BN ^ 2 + AN ^ 2 = AB ^ 2;
(3R / 2) ^ 2 + (a / 2) ^ 2 = a ^ 2;
9R ^ 2 + a ^ 2 = 4a ^ 2;
3a ^ 2 = 9R ^ 2;
a ^ 2 = 3R ^ 2;
a = R√3.
4. To the right-angled triangle A1AC again apply the Pythagorean theorem:

H ^ 2 + a ^ 2 = L ^ 2;
H ^ 2 = L ^ 2 – a ^ 2 = L ^ 2 – (R√3) ^ 2 = L ^ 2 – 3R ^ 2;
H ^ 2 = (√507) ^ 2 – 3 * 11 ^ 2 = 507 – 3 * 121 = 507 – 363 = 144;
H = √144 = 12.
Answer: 12.