Question

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

Answer 1

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.

Complete step by step answer:

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 $${}^{\text{n}}{\text{C}}_r={\displaystyle \frac{n!}{r!\left(n-r\right)!}}$$
So we have
$${}^{\text{n}}{\text{C}}_{\text{2}}\text{ = }{\displaystyle \frac{\text{n!}}{\text{2!}\left(\text{n - 2}\right)\text{!}}}$$
On simplifying we get,
$${\Rightarrow }^{\text{n}}{\text{C}}_{\text{2}}\text{ = }{\displaystyle \frac{\text{n(n - 1)(n - 2)!}}{\text{2!}\left(\text{n - 2}\right)\text{!}}}$$
$$\Rightarrow {\displaystyle \frac{\text{n(n - 1)}}{\text{2}}}$$

Hence, out of the total selections here n are the sides of the polygon so subtracting that from the total selections, we get,
$$\Rightarrow {\displaystyle \frac{\text{n(n - 1)}}{\text{2}}}-n$$
On simplifying we get,
$$\Rightarrow {\displaystyle \frac{\text{n(n - 1) - 2n}}{\text{2}}}$$
On taking n common from both the terms we get,
$$\begin{gathered} \Rightarrow {\displaystyle \frac{\text{n(n - 1 - 2)}}{\text{2}}} \\ \Rightarrow {\displaystyle \frac{\text{n(n - 3)}}{\text{2}}} \end{gathered}$$
Hence , there are total $${\displaystyle \frac{\text{n(n - 3)}}{\text{2}}}$$ 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.