2019 MPFG Problem 17

Problem

Let $P$ be a right prism whose two bases are equilateral triangles with side length $2$ . The height of $P$ is $2\sqrt{3}$ . Let $l$ be the line connecting the centroids of the bases. Remove the solid, keeping only the bases. Rotate one of the bases $180^\circ$ about $l$ . Let $T$ be the convex hull of the two current triangles. What is the volume of $T$ ?

Solution 1

Here is a demonstration of the actual transformation

[[File::2019MPFG 17.jpg|450px|center]]

As we can see, the transformation creates a rectangular prism with $4$ triangular pyramids cut off from the corners.

The volume of the rectangular prism is $2 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3} = 12$

Subtract the volume of the $4$ triangular pyramids, and we get: $V = 12 - 4 \cdot \frac{1}{2} \cdot 1 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3}$ $= 12 - 8 = \boxed{4}$

~cassphe

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2019 MPFG Problem 17

Problem

Solution 1