1989 AMC 12 Problem 26
Attempt Problem 26 of the 1989 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1989 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
26.
A regular octahedron is formed by joining the centers of adjoining faces of a cube. The ratio of the volume of the octahedron to the volume of the cube is
Answer: C
Small hint:
Scale the cube to side length and place its center at the origin
Big hint:
The octahedron has vertices
Solution:
Take a cube of side so its volume is The face centers are In each octant, the octahedron cuts out a tetrahedron with three perpendicular unit edges and volume Thus its volume is and the ratio is
Thus the correct answer is C.
Problem 26 in other years
- 1950 AMC 12
- 1951 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
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