How do you find how many diagonals in a shape?

Answer 1

That's easy enough. In any n-sided polygon, each of the n verticies is the endpoint of n - 3 diagonals, because any of the other endpoints is a different vertex on the polygon, other than the the opposite endpoints on the sides to which the vertex belongs.

At the same time, if you do this, all the diagonals are counted twice, because segment AB is the same as segment BA. Therefore, you divide by 2 and get n(n - 3)/2.

So, for example, consider the pentagon. Each of the five vertices has two others with which a diagonal could be drawn, because there's four other vertices and two already have this one for a side. So five times two is ten, but each was counted twice, so we divide by two, and half of ten is five, the correct answer.