Find the area of a sector of a circle of radius 28 cm and central angle \(45^\circ\).
\(98\pi\;\text{cm}^2\) (≈ 307.88 cm²)
Step 1: Recall the formula
The area of a sector of a circle is given by:
\[
A = \dfrac{\theta}{360^\circ} \times \pi r^2
\]
where:
• \(\theta\) = central angle in degrees
• \(r\) = radius of the circle
• \(\pi \approx 3.1416\)
Step 2: Write the given values
• Radius, \(r = 28\;\text{cm}\)
• Central angle, \(\theta = 45^\circ\)
Step 3: Substitute into the formula
\[
A = \dfrac{45}{360} \times \pi \times (28)^2
\]
Step 4: Simplify the fraction
\[
\dfrac{45}{360} = \dfrac{1}{8}
\]
So,
\[
A = \dfrac{1}{8} \times \pi \times (28)^2
\]
Step 5: Square the radius
\[
(28)^2 = 784
\]
Therefore,
\[
A = \dfrac{1}{8} \times \pi \times 784
\]
Step 6: Divide 784 by 8
\[
\dfrac{784}{8} = 98
\]
So,
\[
A = 98\pi \; \text{cm}^2
\]
Step 7: Approximate the value of \(\pi\)
\[
98 \times 3.1416 \approx 307.88
\]
Hence,
\[
A \approx 307.88\;\text{cm}^2
\]
Final Answer:
The area of the sector is \(98\pi\;\text{cm}^2\) (≈ 307.88 cm²).