Thread: volume of a cube using space diagonals

so i believe the volume of a cube using space diagonals is

V = d^3 / 3sqrt(3) where d is the space diagonal

can anyone help me prove this? the question this relates to is actually the volume of a rectangular prism in terms of space diagonal, but to understand the cube scenario would be a good starter.

apologies for not using latex, i havent used it in a while and need to re-educate myself

thankyou in advance

Yes. Using a 1by1by1 cube: space diagonal = sqrt(3)
From a right triangle 1-sqrt(2)-sqrt(3)

Hello, walleye!

. . . Right!

We have a cube with side

Code:

             x
         *--------C
        /       /|
       /       / | x
      *-------*  |
      |       |  *B
    x |       | /
      |       |/ x
     A*-------*
          x

Consider the "floor" of the cube.

Code:

      *-------* B
      |     * |
      |   *   | x
      | *     |
    A *-------*
          x

We see that: .



Now "slice" the cube through vertices and

We have this cross-section:

Code:

                  * C
           d   *  |
            *     | x
         *        |
    A *-----------* B
            _
          x√2

And we have: .

. .


The volume of the cube is