Q: how the heck long is an interior diagonal line of a cube
A: let's see..... leg squared plus leg squared equals hypotenuse squared...... so let's start with the square face... call the edge 1. 1+1= hypotenuse squared, so the hypotenuse diagonal across the face is √2. Then i can imagine another right triangle with that hypotenuse, the outer edge of the cube, and the grand diagonal (the one you're looking for). so it should be 1 squared plus √2 squared equals answer squared.... 1 plus (root 2 squared)  is 3.... the answer should be the square root of 3. If one edge of the cube is 1, then the grand diagonal should be the square root of 3