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.