Let's break down the problem step by step to calculate the area of a regular polygon given its exterior angle and side length.
Step 1: Calculate the number of sides
The exterior angle of a regular polygon is related to the number of sides [tex]\( n \)[/tex] by the formula:
[tex]\[
\text{Exterior angle} = \frac{360^\circ}{n}
\][/tex]
Given that the exterior angle is [tex]\( 18^\circ \)[/tex]:
[tex]\[
18 = \frac{360}{n}
\][/tex]
Solving for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{360}{18} = 20
\][/tex]
So, the polygon has 20 sides.
Step 2: Calculate the interior angle
The interior angle of a regular polygon is given by:
[tex]\[
\text{Interior angle} = 180^\circ - \text{Exterior angle}
\][/tex]
Given the exterior angle is [tex]\( 18^\circ \)[/tex]:
[tex]\[
\text{Interior angle} = 180^\circ - 18^\circ = 162^\circ
\][/tex]
Step 3: Calculate the area
The formula for the area [tex]\( A \)[/tex] of a regular polygon with [tex]\( n \)[/tex] sides, each of length [tex]\( s \)[/tex], is:
[tex]\[
A = \frac{n s^2}{4 \tan\left(\frac{\pi}{n}\right)}
\][/tex]
Given:
- [tex]\( n = 20 \)[/tex]
- [tex]\( s = 16 \)[/tex] cm
Now, plug these values into the formula:
[tex]\[
A = \frac{20 \times 16^2}{4 \times \tan\left(\frac{\pi}{20}\right)}
\][/tex]
Calculate [tex]\( 16^2 \)[/tex]:
[tex]\[
16^2 = 256
\][/tex]
So the area formula becomes:
[tex]\[
A = \frac{20 \times 256}{4 \times \tan\left(\frac{\pi}{20}\right)}
\][/tex]
Calculate the denominator:
[tex]\[
4 \times \tan\left(\frac{\pi}{20}\right) \approx 4 \times 0.1584 \approx 0.6336
\][/tex]
Now, calculate the numerator:
[tex]\[
20 \times 256 = 5120
\][/tex]
Finally, calculate the area:
[tex]\[
A \approx \frac{5120}{0.6336} \approx 8081.6 \text{ cm}^2
\][/tex]
Therefore, the area of the polygon is approximately [tex]\( 8081.60 \text{ cm}^2 \)[/tex].
In summary:
- The polygon has 20 sides.
- The interior angle is [tex]\( 162^\circ \)[/tex].
- The area of the polygon is approximately [tex]\( 8081.60 \)[/tex] cm[tex]\(^2\)[/tex] to 2 decimal places.
