1. ## Spatial geometry

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.

2. 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 $x$ = edge of the cube.

Side view:
Code:
      : - - - x - - - :
- *-------*-------*
: |     * | *     |
: |   *   |   *   | ½x
: | *     |     * |
x * - - - + - - - * A
: | *     |     * |
: |   *   |   *   | ½x
: |     * | *     |
- *-------*-------* C
B   ½x

From right triangle $ABC$, the edge of the octahedran is: . $AB \:=\:\frac{x}{\sqrt{2}}$

The octahedron is comprised of two pyramids with square bases.
The square base has side $\frac{x}{\sqrt{2}}$; its area is: $B = \frac{x^2}{2}$
. . and its height is $\frac{x}{2}$

The volume of the pyramid is: . $\frac{1}{3}Bh \:=\:\frac{1}{3}\left(\frac{x^2}{2}\right)\left(\f rac{x}{2}\right) \;=\;\frac{x^3}{12}$

Therefore, the volume of the octahedron is: . $2 \times \frac{x^3}{12} \:=\:\frac{x^3}{6}$

3. Originally Posted by Soroban
Hello, Apprentice123!

Let $x$ = edge of the cube.

Side view:
Code:
      : - - - x - - - :
- *-------*-------*
: |     * | *     |
: |   *   |   *   | ½x
: | *     |     * |
x * - - - + - - - * A
: | *     |     * |
: |   *   |   *   | ½x
: |     * | *     |
- *-------*-------* C
B   ½x

From right triangle $ABC$, the edge of the octahedran is: . $AB \:=\:\frac{x}{\sqrt{2}}$

The octahedron is comprised of two pyramids with square bases.
The square base has side $\frac{x}{\sqrt{2}}$; its area is: $B = \frac{x^2}{2}$
. . and its height is $\frac{x}{2}$

The volume of the pyramid is: . $\frac{1}{3}Bh \:=\:\frac{1}{3}\left(\frac{x^2}{2}\right)\left(\f rac{x}{2}\right) \;=\;\frac{x^3}{12}$

Therefore, the volume of the octahedron is: . $2 \times \frac{x^3}{12} \:=\:\frac{x^3}{6}$

Thank you