https://file.aiquickdraw.com/mathbot/file/1725905523213psv0f2tc.jpg

The image presents a regular polygon, specifically an octagon, with a radius of 8 units. We are tasked with finding the area of this octagon.

### Step-by-step solution:

1. **Number of sides (n):** An octagon has 8 sides.
   
2. **Radius (r):** The distance from the center to a vertex, given as 8 units.

3. **Formula for the area of a regular polygon:**
   The formula for the area \( A \) of a regular polygon is:
   \[
   A = \frac{1}{2} n r^2 \sin\left(\frac{2\pi}{n}\right)
   \]
   where:
   - \( n \) is the number of sides,
   - \( r \) is the radius,
   - \( \sin\left(\frac{2\pi}{n}\right) \) is the sine of the central angle.

4. **Substitute values:**
   \[
   A = \frac{1}{2} \times 8 \times 8^2 \times \sin\left(\frac{2\pi}{8}\right)
   \]
   Simplify:
   \[
   A = 4 \times 64 \times \sin\left(\frac{\pi}{4}\right)
   \]
   Since \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \):
   \[
   A = 256 \times \frac{\sqrt{2}}{2}
   \]

5. **Final computation:**
   \[
   A \approx 256 \times 0.707 = 181.81
   \]

Rounding to the nearest whole number, the area is approximately **182 square units**.

Would you like more details on any step or have further questions?

### Related questions:
1. How does the formula for the area of a polygon differ for other shapes?
2. What is the formula for the perimeter of a regular polygon?
3. Can the area formula be applied to polygons with more sides, like a dodecagon?
4. What happens to the area if the radius doubles?
5. How would you find the area of a regular polygon if only the side length is known?

### Tip:
For regular polygons, as the number of sides increases, the shape becomes closer to a circle, and you can approximate its area using the formula for the area of a circle.

Find the area of the regular polygon. Round your answer to the nearest whole number of square units.

Learn how to calculate the area of a regular octagon with a given radius of 8 units. This step-by-step guide explains the use of polygon area formulas and trigonometric functions, suitable for Grades 9-11.

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