Question

The measure of each interior angle of a regular polygon is eleven times that of an exterior angle. How many sides does the polygon have?

Short answer

The polygon has 12 sides.

Step-by-step solution

Step 1. Write the formula to find the measure of each interior angle of the regular polygon.

The formula to find the measure of each interior angle of the regular polygon with sides is $\frac{\left(n-2\right)180}{n}$.

Step 2. Write the formula to find the measure of each exterior angle of the regular polygon.

The formula to find the measure of each exterior angle of the regular polygon with sides is $\frac{360}{n}$.

Step 3. Find the equation for the given statement.

Consider the number of sides of the polygon be $n$.

As, the measure of each interior angle of a regular polygon is elven times that of an exterior angle.

Therefore, the equation for the given statement is:

$$\begin{gathered} \text{measureofeachinteriorangle}=11\times \text{measureofeachexteriorangle} \\ \frac{\left(n-2\right)180}{n}=11\times \frac{360}{n} \end{gathered}$$

Step 4. Find the value of n.

Solve the above equation to find the value of n.

$$\begin{gathered} \frac{\left(n-2\right)180}{n}=11\times \frac{360}{n} \\ \left(n-2\right)180=11\times 360 \\ n-2=\frac{11\times 360}{180} \\ n-2=11\times 2 \\ n-2=22 \\ n=22+2 \\ n=24 \end{gathered}$$

Therefore, the value of n is 24.