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To find the area of a regular octagon given its radius (distance from the center to a vertex), we can use the following method:

1. **Determine the side length (s) of the octagon** using the radius (r). For a regular octagon:
\[ s = r \cdot \sqrt{2 - 2 \cos\left(\frac{360^\circ}{8}\right)} = r \cdot \sqrt{2 - 2 \cos(45^\circ)} \]
Since \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\):
\[ s = 9 \cdot \sqrt{2 - 2 \cdot \frac{\sqrt{2}}{2}} = 9 \cdot \sqrt{2 - \sqrt{2}} \]

2. **Calculate the area of the octagon** using the formula for the area of a regular octagon:
\[ \text{Area} = 2 \cdot (1 + \sqrt{2}) \cdot s^2 \]
First, let's find \(s\) approximately:
\[ s \approx 9 \cdot \sqrt{2 - \sqrt{2}} \approx 9 \cdot \sqrt{2 - 1.414} \approx 9 \cdot \sqrt{0.586} \approx 9 \cdot 0.765 = 6.885 \]

3. **Substitute \(s\) back into the area formula**:
\[ \text{Area} = 2 \cdot (1 + \sqrt{2}) \cdot (6.885)^2 \approx 2 \cdot (1 + 1.414) \cdot 47.41 \approx 2 \cdot 2.414 \cdot 47.41 \approx 4.828 \cdot 47.41 \approx 228.99 \]

Thus, the area of the octagon is approximately 229.0 square units (rounded to the nearest tenth).

Would you like more details or have any questions? Here are some related questions to consider:

1. How do you derive the formula for the side length of a regular polygon given its radius?
2. What is the general formula for the area of a regular n-sided polygon?
3. How does the cosine function relate to the calculation of side lengths in regular polygons?
4. What are some real-life applications of calculating the area of an octagon?
5. How would the formula change if you were given the side length instead of the radius?

**Tip:** When working with trigonometric functions in geometry, always check if you need to convert degrees to radians or vice versa depending on the context of your problem.

O is the center of the regular octagon below. Find its area. Round to the nearest tenth if necessary.

Learn how to calculate the area of a regular octagon given a radius of 9 units. This geometry problem uses trigonometric concepts and the formula for the area of a regular polygon. Suitable for high school students in Grades 9-11.

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