Let's solve the problem step by step.
### Step-by-step Solution
1. Define the Polygon:
- We are working with a regular polygon with 10 sides, meaning all sides and all angles are equal.
- Each side of the polygon measures 2 inches.
2. Calculate the Perimeter:
- The perimeter of a polygon is the total length around the polygon.
- Since our polygon has 10 sides and each side measures 2 inches, we can calculate the perimeter as follows:
[tex]\[
\text{Perimeter} = \text{number of sides} \times \text{side length}
\][/tex]
Substituting in the given values:
[tex]\[
\text{Perimeter} = 10 \times 2 = 20 \text{ inches}
\][/tex]
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]\(a\)[/tex] is:
[tex]\[
\text{Area} = \frac{1}{4} \times n \times a^2 \times \cot\left(\frac{\pi}{n}\right)
\][/tex]
- In this problem, [tex]\(n = 10\)[/tex] and [tex]\(a = 2 \text{ inches}\)[/tex]. Plugging in these values:
[tex]\[
\text{Area} = \frac{1}{4} \times 10 \times (2)^2 \times \cot\left(\frac{\pi}{10}\right)
\][/tex]
- Without going into detailed trigonometric computation, it's given that:
[tex]\[
\text{Area} \approx 30.77683537175254 \text{ square inches}
\][/tex]
### Conclusion
- Perimeter of the Polygon: 20 inches
- Area of the Polygon: Approximately 30.7768 square inches
This completes the solution for the problem of drawing and calculating the properties of a 10-sided polygon with each side measuring 2 inches.
