Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Find the answer

16. Given that a sphere is tangent to the three sides and two bases of a regular triangular prism. If the volume of the sphere is $\frac{32 \pi}{3}$ , then the volume of the triangular prism is $\qquad$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

$16.48 \sqrt{3}$ .
From the volume, the radius of the sphere $R=2$ , the height of the triangular prism is 4, and the side length of the base is $4 \sqrt{3}$ .
Thus, $V=\frac{\sqrt{3}}{4}(4 \sqrt{3})^{2} \times 4=48 \sqrt{3}$ .

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