Question

How many diagonals are there in a 50-sided polygon?

Answer 1

`1175`

Explanation:

The number of diagonals of an n-sided polygon is:

`(n(n - 3)) / 2`

It is very immediate to understand: from any vertex, you can draw diagonals to every other vertex, except three: the vertex itself, and the one immediately before and after.

So, for each of the `n` vertices you have `n-3` choices, for a total of `n(n-3)`

Nevertheless, you're counting each diagonal twice in this process, so you divide by two to get to the final formula.

So, you have

`\frac{50(50-3)}{2} = \frac{cancel(50)\cdot 47}{cancel(2)} = 25*47 = 1175`