If a polygon of n sides has 560 diagonals, then n =

The number of segments obtained by joining $n$ vertices of a $n$-sided polygon will be ${}^{n}C_2$ out of which $n$ will be the sides of the polygon.

So, the number of diagonals will be ${}^{n}C_2-n$.

i.e. ${}^{n}C_2-n=560$

$\Rightarrow \frac{n\left(n-1\right)}{2}-n=560\Rightarrow n^2-3n-1120=0$

$\Rightarrow \left(n+32\right)\left(n-35\right)=0$

Hence, $n=35$