Question

The interior angles of a polygon are in A.P. If the smallest angle is $120^\circ$ and the common difference is $5^\circ$, then the number of sides in the polygon is

• A. 7
• B. 9
• C. 16
• D. none of these

Answer 1

Answer: (B)

9

Answer 2

Step 1: Sum of interior angles.
For an $n$-sided polygon,

$$\text{Sum} = (n-2)\times180^\circ.$$

Step 2: Sum via AP.
The interior angles are in AP with first term $a = 120^\circ$ and common difference $d = 5^\circ$. So

$$\text{Sum} = \frac{n}{2}\bigl[2a + (n-1)d\bigr] = \frac{n}{2}\bigl[2\times120^\circ + (n-1)\times5^\circ\bigr].$$

Step 3: Equate sums.

$$(n-2)\times180^\circ \;=\;\frac{n}{2}\bigl[240^\circ + 5(n-1)\bigr] \;\Longrightarrow\;(n-2)\times360 = n(5n+235).$$

This simplifies to

$$n^2 -25n +144 = 0 \;\implies\;(n-16)(n-9)=0 \;\implies\;n=16\text{ or }9.$$

Step 4: Validity check.
The largest angle must be $<180^\circ$:

$$a+(n-1)d <180^\circ \;\implies\;120^\circ +5(n-1)<180^\circ \;\implies\;n<13.$$

Hence $n=9$ is the only feasible solution.