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The formula for finding the number of diagonals in a n-sided convex polygon is:

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

But how is this formula derived? How would I want to start deriving this formula?

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2 Answers

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For each vertex $n $(vertices) you have $(n-3)$ possible vertices to create a diagonal from.

Why $(n-3)$? Because you can't combine with the 2 neighboring vertices and you can't combine with your current vertex. Divide by two because you are counting twice. So you have $\frac{(n-3)n}{2}$ diagonals.

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We can solve this by using combination, to make diagonals we need to choose no. of pairs of 2 vertices that can be formed from $n$ vertices i.e $$ \dbinom{n}{2}$$ we will get the answer, but we need to subtract $n$ from it since adjacent vertices cannot make a diagonal. $\therefore$ the final answer is $$\dbinom{n}{2}-n= \dfrac{(n-1)n-2n}{2}=\dfrac{n(n-3)}{2}.$$

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