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Math Help - volume of a cube using space diagonals

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

    \text{I believe the volume of a cube using space diagonals is:}

    . . V \:=\:\dfrac{d^3}{3\sqrt{3}}\;\text{ where }d\text{ is the space diagonal.} . Right!

    We have a cube with side \,x.


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



    Consider the "floor" of the cube.

    Code:
          *-------* B
          |     * |
          |   *   | x
          | *     |
        A *-------*
              x

    We see that: . AB^2 \:=\:x^2+x^2 \quad\Rightarrow\quad AB = x\sqrt{2}



    Now "slice" the cube through vertices A, B, and C.

    We have this cross-section:

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

    And we have: . (x\sqrt{2})^2 + x^2 \:=\:d^2 \quad\Rightarrow\quad 2x^2 + x^2 \:=\:d^2

    . . 3x^2 \:=\:d^2 \quad\Rightarrow\quad x^2 \:=\:\dfrac{d^2}{3} \quad\Rightarrow\quad x \:=\:\dfrac{d}{\sqrt{3}}


    The volume of the cube is \,x^3

    \text{Therefore: }\;V \;=\;\left(\dfrac{d}{\sqrt{3}}\right)^3 \;=\;\dfrac{d^3}{3\sqrt{3}}

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