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 \[^{\text{n}}{{\text{C}}_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
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.