The Pythagorean theorem is useful when we need to find the length of a space diagonal in a rectangular prism. The space diagonal is the diagonal passing through the space between vertices, instead of the bases or lateral faces. In other words, it is the diagonal that joins two opposing vertices in the prism, or the two vertices that don't share a face with each other.
One application of the Pythagorean Theorem is the space diagonal which is not ride at Disney land. A space diagonal, is the diagonal that passes through space between vertices not along the bases or the lateral faces. So what that means is that if you have a box here, the space diagonal will go from 1 vertex through the middle of that box all the way to its opposite corner. So there's going to be 4 of these and I'll draw in one of them, let's say we started at this vertex right here which is in the upper left. The space diagonal will go all the way down to this lower right vertex. So if I drew that in it's going to look something like that. So in order to calculate the length of this space diagonal, you're going to use the Pythagorean Theorem and you're going to draw in a diagonal along the base of this face. So if you know that this is a right angle, which it will be because, these problems will only apply to rectangular prisms.
Then you find one leg, find the other leg and then use Pythagorean Theorem a squared plus b squared equals c squared to find the length of your space diagonal. So there's 3 other ones and that would extend from this vertex to the one in the front left and you could have one from this back left extending to the front right and finally from this back right extending to the front left. So the key to the problems with space diagonals is remembering that you can apply the Pythagorean Theorem, usually you're going to have apply it more than once.