A straight prism is inscribed in the cylinder, at the base of which is a triangle with sides of 6 cm and 6 cm

| August 22, 2021 | education

A straight prism is inscribed in the cylinder, at the base of which is a triangle with sides of 6 cm and 6 cm and an angle of 120 degrees between them. Find the volume of the cylinder if the axial section of the cylinder is square.

From the triangle ABC, which lies at the base of the prism, by the cosine theorem, we determine the length of the AC side.

AC ^ 2 = AB ^ 2 + BC ^ 2 – 2 * AB * BC * Cos120 = 36 + 36 – 2 * 6 * 6 * (-1 / 2) = 72 + 36 = 108.

AC = 6 * √3 cm.

Determine the area of ​​the triangle ABC. Savs = (AB * BC * Sin120) / 2 = (6 * 6 * √3 / 2) / 2 = 9 * √3 cm2.

Determine the radius of the circumscribed circle around the triangle ABC.

R = ОА = (АВ * ВС * АС) / 4 * Saс = (6 * 6 * 6 * √3) / (4 * 9 * √3) = 6 cm.

Determine the area of ​​the circle at the base of the cylinder.

Sosn = n * OA ^ 2 = 36 * n cm2.

Since, by condition, the diagonal section of the cylinder is a square, the height of the cylinder is equal to the diameter of the circle.

AA1 = 2 * OA = 12 cm.

Determine the volume of the cylinder.

V = Sosn * AA1 = 36 * n * 12 = 432 * n cm3.

Answer: The volume of the cylinder is 432 * n cm3.



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