The Longest Diagonal of a Cube is this guy.

If You Want to Skip the Backstory

longest diagonal of a cube $=\sqrt{3}x$

I Want the Backstory. How Do We Find This Guy?

Imagine we have a right triangle, outlined in red.

We know the height of this right triangle is $x$. If we find the base, we can then do Pythagorean theorem to find the hypotenuse, which just so happens to be the longest diagonal of the cube.

But How Do We Find the Base?

Imagine another right triangle, this time on the bottom of the cube. Notice how the base is the hypotenuse of a $45$-$45$-$90$ triangle, which is equal to $\sqrt{2}x$.

Put It All Together

Now that we have our base, we can solve another Pythagorean theorem problem to get the longest diagonal of the cube.

$$(\sqrt{2}x)^2 + x^2 = hypotenuse^2$$

$$2x^2 + x^2 = hypotenuse^2$$

$$3x^2 = hypotenuse^2$$

$$\sqrt{3}x = hypotenuse$$