To determine the area of a regular nonagon (a nine-sided polygon) with side lengths of six inches each, we use a specific area formula for regular polygons, particularly nonagons.
The formula for the area [tex]\( A \)[/tex] of a regular nonagon with side length [tex]\( s \)[/tex] is:
[tex]\[ A = \frac{9}{4} \times s^2 \times \frac{1}{\tan\left(\frac{\pi}{9}\right)} \][/tex]
Step-by-Step Solution:
1. Identify the side length and number of sides of the nonagon:
- Side length, [tex]\( s = 6 \)[/tex] inches
- Number of sides, [tex]\( n = 9 \)[/tex]
2. Plug these values into the formula:
[tex]\[
A = \frac{9}{4} \times 6^2 \times \frac{1}{\tan\left(\frac{\pi}{9}\right)}
\][/tex]
3. Calculate [tex]\( 6^2 \)[/tex]:
[tex]\[
6^2 = 36
\][/tex]
4. Multiply by [tex]\( \frac{9}{4} \)[/tex]:
[tex]\[
\frac{9}{4} \times 36 = 81
\][/tex]
5. Evaluate [tex]\( \tan\left(\frac{\pi}{9}\right) \)[/tex]:
[tex]\[
\tan\left(\frac{\pi}{9}\right)
\][/tex]
6. Divide 1 by [tex]\( \tan\left(\frac{\pi}{9}\right) \)[/tex]:
[tex]\[
\frac{1}{\tan\left(\frac{\pi}{9}\right)}
\][/tex]
7. Multiply the results:
[tex]\[
81 \times \frac{1}{\tan\left(\frac{\pi}{9}\right)} = 222.54567097582444 \text{ square inches}
\][/tex]
8. Round the area to the nearest tenth:
[tex]\[
222.5 \text{ square inches}
\][/tex]
Therefore, the area of the mirror, rounded to the nearest tenth of a square inch, is [tex]\( 222.5 \)[/tex] square inches.
The correct answer is:
D. 222.5 in [tex]\( ^2 \)[/tex]
