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 = edge of the cube.
Side view:Code:: - - - x - - - : - *-------*-------* : | * | * | : | * | * | ½x : | * | * | x * - - - + - - - * A : | * | * | : | * | * | ½x : | * | * | - *-------*-------* C B ½x
From right triangle , the edge of the octahedran is: .
The octahedron is comprised of two pyramids with square bases.
The square base has side ; its area is:
. . and its height is
The volume of the pyramid is: .
Therefore, the volume of the octahedron is: .