P: The centers of the faces of the right rectangular prism shown below are joined to create an octahedron. What is the volume of this octahedron?
I've been stuck on the above problem for some time. The first problem for me was visualizing the actual octahedron. First, I drew a cube with the center of each face (located by the intersection of the two diagonals). Then, I attempted connecting said centers. As you can imagine, the diagram appeared very confusing, and was very hard to visualize.
As a result, I reasoned to replace the cube with 2 planes, instead. One vertical, and one completely horizontal plane, which are perpendicular to each other. Each of these planes contain all 6 centers.
Afterwards, I connected all these points, and was finally able to visualize the Octahedron! I then reasoned, that since the two planes are perpendicular to each other, the Octahedron is essentially a three-dimensional rhombus, and thus the volume of the Octahedron is actually the sum of 2 congruent "pyramids" (congruent, since the diagonals of a rhombus bisect each other, and are perpendicular to each other). However, at this point, I'm stuck. I'm not sure how to take advantage of this newfound visualization to find the volume of the pyramids.
Q: How can I find the volume of the Octahedron, given the fact that I know it's composed of 2 congruent pyramids of base length 3, height of 2, and a depth of 5/2 (If I'm correct).
Best Answer
If the given box were $[{-1},1]^3$ the octahedron would be regular and consist of $8$ simplices of volume ${1\over6}\cdot 1^3={1\over6}$, hence have volume ${4\over3}$. In order to obtain the volume in the case at hand multiply with ${3\cdot 4\cdot 5\over 2^3}$, giving the end result $10$.