# Number of Diagonals in a Polygon - Problem 1

The formula for calculating the number of diagonals is:

where "diag" is the number of diagonals, and n is the number of vertices (remember, the number of vertices of a polygon is the same as the number of sides).

For example, find the number of diagonals in a polygon with 22 sides. We know that this polygon has 22 vertices. Simply plug this into the formula.

So, a polygon with 22 sides has 209 diagonals.

We can use the formula for calculating the number of diagonals and apply it to problems. First type of problem that I can guarantee you’re going see is how many diagonals are in a 22-gon or your teacher might make it a little simple and say hexagon. But I’m going to assume that he likes to make you struggle with your homework, so how many are in a 22-gon?

Well let’s start by writing out formula, the number of diagonals and again I’m going to abbreviate ‘diagonal’, diag. is equal to the number of vertices in the polygon times n minus 3. So what this -3 does is it takes out that vertex that you’re looking at and then two consecutive and then we have to divide this by 2 because we don’t want to double count those with diagonals.

So what is n in this problem? N is 22, so I’m going to write n equals 22. A great problem solving strategy in geometry, start with your formula, write down what you’re given, substitute and solve.

So I’m going to substitute in 22, so what I’m going to do is I’m going to erase this and write it over here so that way I can work vertically. So the number of diagonals is equal to 22 times 22 minus 3 is 19, all divided by 2. Now the way that I would do this in my head is I’d say that 22 divided 2 is 11 and 11 times 19, well that’s 11 less than 220. So our number of diagonals is 209. So I’m going to write 209 diagonals as my answer.

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