A composite of a cube and an octahedron

Geometry Level pending

A cube and an octahedron are superimposed to create the polyhedron depicted in the figure above.

If the side length of the cube is 1 1 , what the volume of this polyhedron ?


The answer is 1.5.

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

David Vreken
Dec 10, 2020

The polyhedron consists of a (blue) cube with sides of 1 1 and 6 6 (yellow) pyramids with a height of 1 1 and a square base with diagonals of 1 1 .

Therefore, the volume is V poly = V cube + 6 V pyr = 1 3 + 6 ⋅ 1 3 ⋅ 1 2 ⋅ 1 ⋅ 1 ⋅ 1 2 = 3 2 = 1.5 V_{\text{poly}} = V_{\text{cube}} + 6V_{\text{pyr}} = 1^3 + 6 \cdot \frac{1}{3} \cdot \frac{1}{2} \cdot 1 \cdot 1 \cdot \frac{1}{2} = \frac{3}{2} = \boxed{1.5} .

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