Problem 14 · 2018 Math Kangaroo
Hard
Spatial & Visual Reasoning
cube-viewsspatial-reasoning
An octahedron is inscribed in a cube with side length 1; the vertices of the octahedron are the midpoints of the faces of the cube. How big is the volume of the octahedron?
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Answer: D — \(\tfrac{1}{6}\)
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Hint 1 of 2
The octahedron's vertices are the centres of the cube's six faces.
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Hint 2 of 2
Split the octahedron into two square pyramids.
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Approach: octahedron from face centres of a unit cube
- The six face centres of a unit cube form a regular octahedron made of two square pyramids.
- Each pyramid has base area 1/2 (the square joining four face centres) and height 1/2.
- Volume = 2 · (1/3 · 1/2 · 1/2) = 1/6.
- So the volume is 1/6.
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