Cube and Octahedron Intersection

Geometry Level pending

The figure above shows a cube and an octahedron having the same center intersecting. The solid formed by their intersection is shown below. The intersection is the region that is inside the cube and inside the octahedron. Its faces are squares and equilateral triangles with the same side length. If that side length is equal to 1 1 , then find the volume of the solid.


The answer is 2.357.

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Hongqi Wang
Dec 12, 2020

V = V c u b e − 8 ⋅ V c o n e = ( 2 ⋅ 2 2 ) 3 − 8 ⋅ 1 3 ⋅ [ 1 2 ( 2 2 ) 2 ] ⋅ 2 2 = 2 2 − 1 3 2 = 5 3 2 \begin{aligned} V &= V_{cube} - 8 \cdot V_{cone} \\ &= \left (2 \cdot \dfrac {\sqrt 2}{2} \right )^3 - 8 \cdot \dfrac 13 \cdot \left [\dfrac 12 \left (\dfrac {\sqrt 2}2 \right )^2 \right ] \cdot \dfrac {\sqrt 2}2 \\ &= 2 \sqrt 2 - \dfrac 13 \sqrt 2 \\ &= \dfrac 53 \sqrt 2 \end{aligned}

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