Question

If the difference between the exterior angle of a \[n\] sided polygon and an \[\left( {n + 1} \right)\] sided polygon is \[12^\circ \], find the value of \[n\].

Answer 1

Hint:
Here, we need to find the value of \[n\], for that we will first find the exterior angle of the polygon with \[n\] sides. Then, we will find the exterior angle of the polygon with \[\left( {n + 1} \right)\] sides. We will then subtract both the exterior angle to obtain the desired equation. Simplifying the equation will give us a quadratic equation, which we can solve further to find the value of \[n\].
Formula Used: We will use the formula of the exterior angle of a polygon, \[\dfrac{{360^\circ }}{s}\], where \[s\] is the number of sides.

Complete step by step solution:
The exterior angle of a polygon with \[s\] sides is given by \[\dfrac{{360^\circ }}{s}\].
Now, we know that the first polygon has \[n\] sides.
Substituting \[n\] for \[s\] in the formula for exterior angle of a polygon, we get
Exterior angle \[ = \dfrac{{360^\circ }}{n}\]
Thus, the measure of the exterior angle of the \[n\] sided polygon is \[\dfrac{{360^\circ }}{n}\].
Next, we know that the second polygon has \[n + 1\] sides.
Substituting \[n + 1\] for \[s\] in the formula for exterior angle of a polygon, we get
Exterior angle \[ = \dfrac{{360^\circ }}{{n + 1}}\]
Thus, the measure of the exterior angle of the \[n + 1\] sided polygon is \[\dfrac{{360^\circ }}{{n + 1}}\].
The difference between the two exterior angles is \[12^\circ \]. Therefore, we get
\[\dfrac{{360^\circ }}{n} - \dfrac{{360^\circ }}{{n + 1}} = 12^\circ \]
Now, we will simplify this equation.
Taking the L.C.M. of the terms, we get
\[ \Rightarrow \dfrac{{360\left( {n + 1} \right) - 360n}}{{n\left( {n + 1} \right)}} = 12\]
Multiplying both sides by \[n\left( {n + 1} \right)\], we get
\[ \Rightarrow 360\left( {n + 1} \right) - 360n = 12n\left( {n + 1} \right)\]
Multiplying the terms of the equation, we get
\[ \Rightarrow 360n + 360 - 360n = 12{n^2} + 12n\]
Adding and subtracting the like terms, we get
\[\begin{array}{l} \Rightarrow 12{n^2} + 12n - 360 = 0\\ \Rightarrow {n^2} + n - 30 = 0\end{array}\]
Here, we get a quadratic equation. We will use factoring to solve the quadratic equation.
Factoring the quadratic equation, we get
\[\begin{array}{l} \Rightarrow {n^2} + 6n - 5n - 30 = 0\\ \Rightarrow n\left( {n + 6} \right) - 5\left( {n + 6} \right) = 0\\ \Rightarrow \left( {n + 6} \right)\left( {n - 5} \right) = 0\end{array}\]
Using zero product rule, we can write above equation as

\[\begin{array}{c}n + 6 = 0\\n = - 6\end{array}\]
or
 \[\begin{array}{c}n - 5 = 0\\n = 5\end{array}\]
We know that the number of sides of a polygon cannot be negative.

Thus, the value of \[n\] is 5.

Note:
We should remember the formula for the exterior angle of a polygon with \[s\] sides to solve this question. Here, we have used the concept of zero product rule to simplify the equation. Zero product rule states that if \[mn = 0\], then either \[m\] is equal to zero or \[n\] is equal to zero. Another thing we need to remember is that the number of sides of a polygon cannot be negative. Therefore, \[n = - 6\] is not acceptable as the answer.