Question

# How many diagonals are there in a polygon with n sides.

Hint: A polygon of n sides should also be having n vertices. By joining any two vertices of a polygon, we obtain either a side of that polygon or a diagonal of that polygon. So calculating with the help of permutation, by taking ${\text{2}}$ points at a time, we get the number of lines joining all the points, then subtracting the number of edges we get the total number of diagonals.

We have to find the numbers of diagonals in the n-sided polygon.
The number of line segments obtained by joining the vertices of a n sided polygon taken two points at a time.
Now, applying the formula and using permutation as below stated.
The number of ways of selecting 2 points at a time from n number of points is given as $^{\text{n}}{{\text{C}}_{\text{2}}}$
As we have $^{\text{n}}{{\text{C}}_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
So we have
$^{\text{n}}{{\text{C}}_{\text{2}}}{\text{ = }}\dfrac{{{\text{n!}}}}{{{\text{2!}}\left( {{\text{n - 2}}} \right){\text{!}}}}$
On simplifying we get,
${ \Rightarrow ^{\text{n}}}{{\text{C}}_{\text{2}}}{\text{ = }}\dfrac{{{\text{n(n - 1)(n - 2)!}}}}{{{\text{2!}}\left( {{\text{n - 2}}} \right){\text{!}}}}$
$\Rightarrow \dfrac{{{\text{n(n - 1)}}}}{{\text{2}}}$

Hence, out of the total selections here n are the sides of the polygon so subtracting that from the total selections, we get,
$\Rightarrow \dfrac{{{\text{n(n - 1)}}}}{{\text{2}}} - n$
On simplifying we get,
$\Rightarrow \dfrac{{{\text{n(n - 1) - 2n}}}}{{\text{2}}}$
On taking n common from both the terms we get,
$\Rightarrow \dfrac{{{\text{n(n - 1 - 2)}}}}{{\text{2}}} \\ \Rightarrow \dfrac{{{\text{n(n - 3)}}}}{{\text{2}}} \\$
Hence , there are total $\dfrac{{{\text{n(n - 3)}}}}{{\text{2}}}$ number of diagonals in an n sided polygon.

Note:: Don’t forget to subtract the number of sides while finding the number of diagonals. In geometry, a polygon is a plane figure that is described by a finite number of straight-line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon. In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal.