Verified Answer :
- Define the radii of the circular boundaries as \(r_1 = 2 \text{ m}\) and \(r_3 = 6 \text{ m}\) and the central angle as \(\theta = 45^{\circ}\).
- Calculate the areas of the sectors corresponding to the larger and smaller circles using the formula \(\frac{\theta}{360^{\circ}} \times \pi r^2\).
- Find the shaded area by subtracting the area of the smaller sector from the area of the larger sector.
- The final calculated area of the region is \(\boxed{13 \text{ m}^2}\).
Explanation
- Define the geometry and the formula for the sector area.
To calculate the area of the shaded region, we define the boundaries using the given radii. The central circle has a radius of \(r_1 = 2 \text{ m}\), and the outermost ring has a radius of \(r_3 = 6 \text{ m}\). We are interested in the area of a sector with a central angle of \(\theta = 45^{\circ}\) that lies between these two circles. The area of a circular sector with radius \(r\) and central angle \(\theta\) is calculated using the formula:
\[A = \frac{\theta}{360^{\circ}} \times \pi \times r^2\]
- Calculate the area of the smaller and larger sectors.
We first calculate the area of the sector of the circle with radius \(r_1 = 2 \text{ m}\):
\[A_1 = \frac{45^{\circ}}{360^{\circ}} \times \pi \times (2 \text{ m})^2 = \frac{1}{8} \times \pi \times 4 \text{ m}^2 = 0.5\pi \text{ m}^2\]
Next, we calculate the area of the sector of the circle with radius \(r_3 = 6 \text{ m}\):
\[A_3 = \frac{45^{\circ}}{360^{\circ}} \times \pi \times (6 \text{ m})^2 = \frac{1}{8} \times \pi \times 36 \text{ m}^2 = 4.5\pi \text{ m}^2\]
- Calculate the shaded area and round the result.
The area of the shaded region is the difference between the area of the larger sector and the area of the smaller sector:
\[A_{shaded} = A_3 - A_1 = 4.5\pi \text{ m}^2 - 0.5\pi \text{ m}^2 = 4\pi \text{ m}^2\]
Using the value of \(\pi \approx 3.14159\):
\[A_{shaded} = 4 \times 3.14159 \text{ m}^2 \approx 12.566 \text{ m}^2\]
Rounding to the nearest square metre, we obtain \(13 \text{ m}^2\). The final area of the shaded region is \(\boxed{13 \text{ m}^2}\).
Examples
In landscape architecture, calculating the area of circular regions is essential for determining how much material (such as mulch, grass seed, or decorative stone) is needed. For instance, if you are planning a curved flower bed or a segment of a circular patio, you can use the sector area formula to find the precise amount of supplies required for your project. This approach helps in budgeting accurately and minimizing waste.