How do I find the length of diagonal line AB in a cube with a volume of 2197mm^3? How can I use the Pythagorean theory here?

We need to find the side length....this will be the cube root of 2197 = 13mm

Since we're in 3-D, we have something that is analogous to the P Theorem for finding the length of the diagonal, D.....it is given by

D = √[s^2 + s^2 + s^2] =    √[3 s^2]  = s√3

So

D = 13√3  = about 22.52 mm

BTW....this "formula" can be developed from using the P Theorem twice.....I'll derive it only if you're interested.....