A regular triangular prism is inscribed in the cylinder with a radius so that the base of the prism

| March 26, 2021 | education

A regular triangular prism is inscribed in the cylinder with a radius so that the base of the prism is inscribed in the base of the cylinder and its lateral edges are the generatrix of the cylinder. the diagonal of the lateral face of the prism forms an angle a with its lateral edge. Find the surface area of the cylinder

Let the side of the triangle at the base of the prism be a cm, the radius of the circle at the base of the cylinder R cm, and the height of the cylinder h cm.

Since the base of the prism is a regular triangle, the radius of the circle described around it will be equal to: R = a * √3 / 3 cm, then a = AC = AB = BC = R * 3 / √3 = R * √3 cm.

In a right-angled triangle ACC1, we determine the length of the leg CC1.

CC1 = h = R * √3 / tgα see.

Determine the circumference at the base of the cylinder. L = 2 * n * R.

Then the area of ​​the lateral surface of the cylinder will be equal to: Side = L * h = 2 * n * R * R * √3 / tanα =

n * 2 * √3 * R ^ 2 / tanα cm2.

Sosn = n * R ^ 2.

S floor = S side + 2 * S main = n * 2 * √3 * R ^ 2 / tanα + 2 * n * R ^ 2 = 2 * n * R ^ 2 * (√3 / tanα + 1) cm2.

Answer: The surface area of ​​the cylinder is 2 * n * R ^ 2 * (√3 / tanα + 1) cm2.



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