Hello, walleye!

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

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

. Right!

We have a cube with side $\displaystyle \,x.$

Code:

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

Consider the "floor" of the cube.

Code:

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

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

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

We have this cross-section:

Code:

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

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

. . $\displaystyle 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 $\displaystyle \,x^3$

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