Question

The perimeter of sector OAB of a circle with centre O and radius \(5.6 \text{ cm}\), is \(15.6 \text{ cm}\). Find length of the arc AB. Also find the value of \(\theta\).

Hint:

Be careful not to confuse "perimeter of sector" with "arc length". The perimeter always includes the two radii (\(OA\) and \(OB\)).

Answer 1

Step 1: Understanding the Concept:
The perimeter of a sector consists of two radii and the arc length.
Step 2: Key Formula or Approach:
- Perimeter = \(2r + l\), where \(l\) is arc length.
- Arc length (\(l\)) = \(\frac{\theta}{360} \times 2 \pi r\)
Step 3: Detailed Explanation:
Given: \(r = 5.6 \text{ cm}\), Perimeter = \(15.6 \text{ cm}\).
1. Find Arc Length (l):
\[ 15.6 = 2(5.6) + l \]
\[ 15.6 = 11.2 + l \]
\[ l = 15.6 - 11.2 = 4.4 \text{ cm} \]
2. Find \(\theta\):
\[ 4.4 = \frac{\theta}{360} \times 2 \times \frac{22}{7} \times 5.6 \]
\[ 4.4 = \frac{\theta}{360} \times 44 \times 0.8 \]
\[ 4.4 = \frac{\theta}{360} \times 35.2 \]
Divide both sides by 4.4:
\[ 1 = \frac{\theta}{360} \times 8 \]
\[ 8 \theta = 360 \]
\[ \theta = \frac{360}{8} = 45^{\circ} \]
Step 4: Final Answer:
The arc length is \(4.4 \text{ cm}\) and the central angle \(\theta\) is \(45^{\circ}\).