Calculate the volume of the octahedron whose edges are the segments connecting the centers of adjacent faces of a cube, in function on the edge of the cube.
Hello, Apprentice123!
Calculate the volume of the octahedron whose edges are the segments connecting
the centers of adjacent faces of a cube, as a function of the edge of the cube.
Let $\displaystyle x$ = edge of the cube.
Side view:Code:: - - - x - - - : - *-------*-------* : | * | * | : | * | * | ½x : | * | * | x * - - - + - - - * A : | * | * | : | * | * | ½x : | * | * | - *-------*-------* C B ½x
From right triangle $\displaystyle ABC$, the edge of the octahedran is: .$\displaystyle AB \:=\:\frac{x}{\sqrt{2}}$
The octahedron is comprised of two pyramids with square bases.
The square base has side $\displaystyle \frac{x}{\sqrt{2}}$; its area is: $\displaystyle B = \frac{x^2}{2}$
. . and its height is $\displaystyle \frac{x}{2}$
The volume of the pyramid is: .$\displaystyle \frac{1}{3}Bh \:=\:\frac{1}{3}\left(\frac{x^2}{2}\right)\left(\f rac{x}{2}\right) \;=\;\frac{x^3}{12}$
Therefore, the volume of the octahedron is: .$\displaystyle 2 \times \frac{x^3}{12} \:=\:\frac{x^3}{6}$