A spider weaved a web in the shape of a regular nonagon. What is the area of the web, to the nearest hundredth, if each side measures 5.4 millimeters?(1 point)

Question

The area of the spiderweb is approximately_____ mm squared solve it correctly
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Answer 1

To calculate the area of a regular nonagon (a polygon with 9 equal sides), we can use the following formula:

\[ \text{Area} = \frac{1}{4} \sqrt{9(5.4^2)} \tan\left(\frac{\pi}{9}\right) \]

Given:

• The number of sides \( n = 9 \)
• The length of each side \( s = 5.4 \) mm

1. First, we calculate \( s^2 \):

\[ s^2 = (5.4)^2 = 29.16 \]

2. Now compute \( 9s^2 \):

\[ 9s^2 = 9 \times 29.16 = 263.44 \]

3. Next, calculate the tangent of \( \frac{\pi}{9} \):

Using a calculator, we find:

\[ \tan\left(\frac{\pi}{9}\right) \approx 0.36397 \]

4. Now substitute these values into the area formula:

\[ \text{Area} = \frac{1}{4} \sqrt{263.44} \tan\left(\frac{\pi}{9}\right) \]

Calculating \( \sqrt{263.44} \):

\[ \sqrt{263.44} \approx 16.24 \]

5. Now plug this into the area equation:

\[ \text{Area} \approx \frac{1}{4} \times 16.24 \times 0.36397 \approx \frac{1}{4} \times 5.9031\approx 1.47578 \]

6. Finally, multiplying by 9:

\[ \text{Area} \approx 9 \times 1.47578 \approx 13.38 \]

So the area of the spiderweb, rounded to the nearest hundredth, is:

\[ \text{Area} \approx 13.38 \text{ mm}^2 \]

Thus, the area of the spider web is approximately 13.38 mm squared.