Problem:
Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Two pyramids with square bases form the octahedron. The upper pyramid is shown.
Since the length of is , the base area of the pyramid is . The altitude of the pyramid is , so its volume is
The volume of the octahedron is .
The problems on this page are the property of the MAA's American Mathematics Competitions