Question

Chord AB of a circle subtends an angle of \(120^\circ\) at the centre O of the circle. Find the length of arc AB, if radius of the circle is 21 cm.

Hint:

Notice that \(\frac{120^\circ}{360^\circ}\) is exactly \(\frac{1}{3}\) of the circle.
The total circumference of the circle is \(2\pi r = 2 \times \frac{22}{7} \times 21 = 132\text{ cm}\).
Dividing 132 by 3 gives 44 cm, which is a fast way to verify your result!

Answer 1

Correct Answer: 44

Step 1: Identify given values.
Radius of circle \(r = 21\) cm, central angle \(\theta = 120^\circ\).
Step 2: Recall the arc length formula.
\[\text{Arc length} = \frac{\theta}{360^\circ} \times 2\pi r\]
Step 3: Substitute the values.
\(l = \frac{120}{360} \times 2 \times \frac{22}{7} \times 21\).
Step 4: Simplify step by step.
\(\frac{120}{360} = \frac{1}{3}\). \(2 \times \frac{22}{7} \times 21 = 2 \times 22 \times 3 = 132\). So \(l = \frac{1}{3} \times 132 = 44\) cm.
Step 5: Verify the setup.
The full circumference \(= 2 \times \frac{22}{7} \times 21 = 132\) cm. One-third of the circle corresponds to a \(120^\circ\) arc, so arc \(= \frac{132}{3} = 44\) cm. Verified!
Step 6: State the final answer.
\[ \boxed{\text{Length of arc AB} = 44 \text{ cm}} \]