Question:
What is the number of diagonals in a
(i) heptagon
(ii) octagon
(iii) polygon of 12 sides?
Solution:
Number of diagonal in an $\mathrm{n}$-sided polygon $=\frac{n(n-3)}{2}$
(i) For a heptagon:
$n=7 \Rightarrow \frac{n(n-3)}{2}=\frac{7(7-3)}{2}=\frac{28}{2}=14$
(ii) For an octagon:
$n=8 \Rightarrow \frac{n(n-3)}{2}=\frac{8(8-3)}{2}=\frac{40}{2}=20$
(iii) For a 12-sided polygon:
$n=12 \Rightarrow \frac{n(n-3)}{2}=\frac{12(12-3)}{2}=\frac{108}{2}=54$