A regular triangular prism is inscribed in the cylinder, the volume of which is V. Find the volume of the cylinder?

Since, according to the condition, the prism is correct, then a regular triangle lies at its base, then AB = BC = AC. Let the side of the triangle be a cm, and the height of the prism h cm.

Since at the base the circle is described around a regular triangle, its radius will be equal to:

R = a / √3 cm, then a = R * √3 cm.

The area of ​​the base of the prism will be equal to: Sbn1 = a2 * √3 / 4.

Then the volume of the prism will be: Vpr = h * a ^ 2 * √3 / 4 = h * (R * √3) ^ 2 * √3 / 4 = h * R ^ 2 * 3 * √3 / 4.

R ^ 2 * h = 4 * Vpr / 3 * √3 = 4 * √3 * Vpr / 9.

The volume of the cylinder is:

Vcyl = n * R ^ 2 * h = n * 4 * √3 * Vpr / 9.

Answer: The volume of the cylinder is n * 4 * √3 * Vpr / 9 cm3.