To find the area of a regular nonagon when given the perimeter, we'll follow these steps:
1. Calculate the Length of One Side: Since the nonagon is regular, all sides are of equal length. The perimeter is the sum of the lengths of all sides. So the length of one side, [tex]\( s \)[/tex], can be calculated by dividing the perimeter, [tex]\( P \)[/tex], by the number of sides, [tex]\( n \)[/tex]. In our case:
[tex]\[ P = 45 \text{ inches} \][/tex]
[tex]\[ n = 9 \][/tex]
So, the length of one side is:
[tex]\[ s = P/n = 45/9 = 5 \text{ inches} \][/tex]
2. Use the Area Formula for a Regular Polygon: The area, [tex]\( A \)[/tex], of a regular polygon can be calculated using the formula:
[tex]\[ A = \frac{n \cdot s^2}{4 \cdot \tan(\pi/n)} \][/tex]
where:
- [tex]\( n \)[/tex] is the number of sides,
- [tex]\( s \)[/tex] is the length of one side, and
- [tex]\( \pi \)[/tex] is approximately 3.14159.
3. Calculate the Area of the Nonagon: Plugging in the values for [tex]\( s \)[/tex] and [tex]\( n \)[/tex], we have:
[tex]\[ A = \frac{9 \cdot 5^2}{4 \cdot \tan(\pi/9)} \][/tex]
Now compute the area:
- First calculate the tangent of [tex]\( \frac{\pi}{9} \)[/tex]:
[tex]\[ \tan(\pi/9) \][/tex]
- Next, compute the area using the value from the previous step:
[tex]\[ A = \frac{9 \cdot 25}{4 \cdot \tan(\pi/9)} \][/tex]
4. Round the Result to the Nearest Hundredth: When you have calculated the area using a calculator, round the number to the nearest hundredth.
Let’s proceed with the calculations step by step. I'll illustrate the process without an actual calculator, but to find the exact number you would use one.
First, we compute the tangent value, for which you would use a calculator:
[tex]\[ \tan(\pi/9) \][/tex]
Then, we plug that value into the formula:
[tex]\[ A = \frac{9 \cdot 25}{4 \cdot \tan(\pi/9)} \][/tex]
Next, you would evaluate this expression using a calculator to get the area in square inches. After computing the expression, you should get the area to a high level of precision. Finally, round off this number to two decimal places to get the area to the nearest hundredth of an inch squared.
For example, if the result of [tex]\( A \)[/tex] were 123.456789 after the calculation, you would round it to give 123.46 square inches. Perform this final step with your actual result to complete the problem.
