A regular triangular prism is inscribed in the cylinder Find. the volume of the prism

| August 5, 2021 | education

A regular triangular prism is inscribed in the cylinder Find. the volume of the prism if the radius of the base of the cylinder is √2 and its generatrix is √3.

Since, by condition, a regular triangle lies at the base of the prism, the central angle AOB will be equal to 120. By the cosine theorem, in a triangle AOB, we determine the length of the side AB.

AB ^ 2 = ОА ^ 2 + ОВ ^ 2 – 2 * ОА * ОВ * Сos120 = 2 + 2 – 2 * √2 * √2 * (-1 / 2) = 4 + 2 = 6.

AB = √6 cm.

Determine the area of an equilateral triangle.

Saws = a ^ 2 * √3 / 4, where a is the length of the side of the triangle.

Saws = 6 * √3 / 4 = 3 * √3 / 2 cm2.

The generatrix of the AD cylinder is equal to the height of the prism OO1, then BE = √2 cm.

Let’s define the volume of the prism.

Vpr = Savs * BE = 3 * √3 / 2 * √3 = 9/2 = 4.5 cm3.

Answer: The volume of the prism is 4.5 cm3.



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