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: Using Perimeter Formula:
Perimeter of sector = 2r + arc length (l)

Given:
r = 5.6 cm
Perimeter = 15.6 cm

15.6 = 2(5.6) + l
15.6 = 11.2 + l
l = 4.4 cm

Step 2: Using Arc Length Formula:
Arc length l = (θ/360) × 2πr

4.4 = (θ/360) × 2 × (22/7) × 5.6

First simplify:
2 × (22/7) × 5.6
= 2 × (22/7) × (56/10)
= 2 × (22 × 8 /10)
= 2 × (176/10)
= 35.2

So,
4.4 = (θ/360) × 35.2

Divide both sides by 4.4:
1 = (θ/360) × 8

8θ = 360
θ = 45°

Final Answer:
Arc length = 4.4 cm
Central angle = 45°