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

May 22, 2018

$1175$

#### Explanation:

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

$\frac{n \left(n - 3\right)}{2}$

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 $n \left(n - 3\right)$

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

So, you have

$\setminus \frac{50 \left(50 - 3\right)}{2} = \setminus \frac{\cancel{50} \setminus \cdot 47}{\cancel{2}} = 25 \cdot 47 = 1175$