To find the area of a regular nonagon (9-sided polygon) with a side length of \( s = 5.4 \) mm, we can use the formula for the area \( A \) of a regular polygon:
\[ A = \frac{1}{4} n s^2 \cot\left(\frac{\pi}{n}\right) \]
where \( n \) is the number of sides and \( s \) is the length of a side.
For a regular nonagon, \( n = 9 \). Plugging the values into the formula gives:
\[ A = \frac{1}{4} \times 9 \times (5.4)^2 \cot\left(\frac{\pi}{9}\right) \]
- Calculate \( (5.4)^2 \):
\[ (5.4)^2 = 29.16 \]
- Now compute the area using the formula:
\[ A = \frac{9}{4} \times 29.16 \times \cot\left(\frac{\pi}{9}\right) \]
- We need to calculate \( \cot\left(\frac{\pi}{9}\right) \). The value of \( \cot\left(\frac{\pi}{9}\right) \) can be found using a calculator or trigonometric tables.
Using a calculator:
\[ \cot\left(\frac{\pi}{9}\right) \approx 3.077 \]
- Now plug this back into the area formula:
\[ A \approx \frac{9}{4} \times 29.16 \times 3.077 \]
- Calculate \( \frac{9}{4} \):
\[ \frac{9}{4} = 2.25 \]
- Now compute:
\[ A \approx 2.25 \times 29.16 \times 3.077 \approx 2.25 \times 89.80932 \approx 202.06 \text{ mm}^2 \]
Finally, rounding to the nearest hundredth:
\[ \text{Area} \approx 202.06 \text{ mm}^2 \]
Thus, the area of the spiderweb is approximately 202.06 mm².
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