I recently reviewed a problem involving a bug on a cylinder. We need to find the shortest distance from the bug’s original position, A, to B which is on the opposite side of the cylinder. The bug can only crawl on the surface, and we’re given the height (8) and circumference (12). One student wanted the bug to crawl along the lower circumference and then crawl straight up the side.
You can more easily visualize the shortest path by using your imagination to “cut” the cylinder along one side starting at point A, then “flatten” the cylinder into a 12×8 rectangle. Now the shortest path is easily seen as the hypotenuse of the triangle with legs of 6 and 8. (A 3-4-5 triangle hiding in plain sight!)