Problem:
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?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Removing the corners removes two segments of equal length from each edge of the cube. Call that length . Then each octagon has side length , and the cube has edge length , so
Each removed corner is a tetrahedron whose altitude is and whose base is an isosceles right triangle with leg length . Thus the total volume of the eight tetrahedra is
The problems on this page are the property of the MAA's American Mathematics Competitions