To find the area of the octagonal park shown in the image, we can decompose the shape into simpler geometric shapes whose areas we can calculate and then sum.

The decomposition of the octagon is as follows:

1. Central rectangle: Dimensions are $12 \, \text{m} \times 8 \, \text{m}$.

   • Area = $12 \times 8 = 96 \, \text{m}^2$.

2. Four trapezoids (on the sides of the rectangle):

   • Each trapezoid has parallel sides $4 \, \text{m}$ and $8 \, \text{m}$, and height $6 \, \text{m}$.
   • Area of one trapezoid = $\frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}$ = $\frac{1}{2} \times (4 + 8) \times 6 = 36 \, \text{m}^2$.
   • Total area of all four trapezoids = $36 \times 4 = 144 \, \text{m}^2$.

3. Two triangles (on the corners):

   • Each triangle is right-angled with legs $4 \, \text{m}$ (height) and $6 \, \text{m}$ (base).
   • Area of one triangle = $\frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 4 \times 6 = 12 \, \text{m}^2$.
   • Total area of both triangles = $12 \times 2 = 24 \, \text{m}^2$.

Total Area:

Adding up the areas of all the components: $\text{Total Area} = 96 + 144 + 24 = 264 \, \text{m}^2.$

Thus, the area of the octagonal park is $264 \, \text{m}^2$.

Would you like me to break it down further or confirm this with another method?

Related Questions:

1. How do you calculate the area of a trapezoid in general?
2. How can we calculate the perimeter of the same octagon?
3. What if the dimensions were doubled? How would the area scale?
4. How is this decomposition strategy useful for irregular shapes?
5. What is the formula for the area of a regular octagon, and how does it differ from this case?

Tip:

For irregular polygons, decomposing them into triangles, rectangles, and trapezoids makes the calculations simpler and more structured.