Problem 340

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Problem 340: (2007 AMC 12a, 20)

Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?

$\mathrm{(A)}\ \frac{5\sqrt{2}-7}{3}\qquad$ $ \mathrm{(B)}\ \frac{10-7\sqrt{2}}{3}\qquad$ $ \mathrm{(C)}\ \frac{3-2\sqrt{2}}{3}\qquad$ $ \mathrm{(D)}\ \frac{8\sqrt{2}-11}{3}\qquad$ $ \mathrm{(E)}\ \frac{6-4\sqrt{2}}{3}$

Solution

Exercise 340: Solution

Corners are sliced off a regular tetrahedron of edge length $3$ by planes that cut each edge at distances $1$ and $2$ from its ends, so that each of the four faces becomes a regular hexagon. What is the volume of the remaining solid?

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