Question

How many diagonals are there in a polygon of $n$ sides?

• A. $\frac{n(n-1)}{2}$
• B. $\frac{n(n-3)}{2}$
• C. $\frac{n(n+1)}{2}$
• D. none of these

Answer 1

Answer: (B)

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

Number of sides $=n$

Number of ends $=n$

For a digonal there will be two end points

Possible digonals

${}^n{C}_2\frac{n(n-1)}{2}$

But these also include the sides of a polygon

now sibstracting $n$ sides

$\Rightarrow \frac{n(n-1)}{2}-n\Rightarrow \frac{n^2-n-2n}{2}\Rightarrow \frac{n^2-3n}{2}\Rightarrow \frac{n(n-3)}{2}$

Hence the correct answer is $\frac{n(n-3)}{2}$