Question

All edges of a triangular prism are of length 8. A cylinder is inscribed in this prism. What is the volume of the inscribed cylinder? Please write your answer as a completely simplified exact value (and in terms of pi whenever applicable).

Answer 1

128π/3 cubic units

Explanation:

You want the volume of a cylinder inscribed in a triangular prism that has all edges 8 units long.

Base

The base of the prism is an equilateral triangle with side length 8. Such a triangle has an altitude of ...

8(√3/2) = 4√3

The centroid of the triangle is also the center of the incircle. That is located 1/3 of the length of the altitude from each side. That distance is the radius of the inscribed cylinder.

r = (4√3)/3 = 4/√3

Then the area of the base of the cylinder is ...

A = πr² = π(4/√3)² = 16π/3

Volume

The height of the cylinder is 8 units, so the volume is ...

V = Bh

V = (16π/3)(8) = 128π/3

The volume of the inscribed cylinder is 128π/3 cubic units.

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