The length of the arc of the sector of a circle with radius 21 cm and of central angle 60°, is : (a) 22 cm (b) 44 cm (c) 88 cm (d) 11 cm

Q: What is the formula for the length of an arc?

Arc length = $(\theta/360°) \times 2\pi r$, where θ is the central angle and r is the radius. It is the corresponding fraction of the full circumference 2πr.

Areas Related to Circles

Length of an arc

Options

(A)22 cm

(B)44 cm

(D)11 cm

Official Answer

(a) 22 cm, obtained from arc length = $(60/360) \times 2 \times (22/7) \times 21 = 22 \text{ cm}$ .

arc lengthsectorcentral angleθ/3602πrπ = 22/7circumference22 cm

Marking Scheme

11 mark for correct option (a) 22 cm.
2Expected working: arc = $(60/360) \times 2 \times (22/7) \times 21 = (1/6) \times 132 = 22 \text{ cm}$ .
3Accept the answer if the correct arc-length formula and $\pi = \frac{22}{7}$ are used.

Hint

Arc length = $(\theta/360) \times 2\pi r$ . Here $\theta/360 = 60/360 = 1/6$ of the circumference.

Quick Oral Answer

Arc length equals theta over 360 times 2 pi r; with 60 degrees, radius 21 and pi as 22 by 7, that is one-sixth of 132, which is 22 centimetres.

Analysis & Explanation

Tests the direct application of the arc-length formula for a sector using the given radius and central angle.

Concept

Arc length of a sector = $\left(\frac{\theta}{360°}\right) \times 2\pi r$ , where θ is the central angle and r is the radius.
Here $\theta = 60°$ , $r = 21 \text{ cm}$ , and $\pi = \frac{22}{7}$ (chosen because 21 is a multiple of 7, simplifying the calculation).

Key points

Substituting gives $\left(\frac{60}{360}\right) \times 2 \times \left(\frac{22}{7}\right) \times 21 = \left(\frac{1}{6}\right) \times 132 = 22 \text{ cm}$ .

Common mistakes

Confusing arc length with sector area (which would involve r² instead of r).
Using π = 3.14 instead of 22/7, which needlessly complicates a problem designed for exact cancellation.

Common Mistakes

1Using the sector area formula $(\theta/360 \times \pi r^2)$ instead of the arc length formula, giving $231 \text{ cm}^2$ and confusion.
2Dropping the factor 2 in $2\pi r$ and getting 11 cm (option d).
3Miscomputing $60/360$ as $1/3$ instead of $1/6$ , doubling the answer to 44 cm.

Interesting Facts

For any circle, a 60° sector always contains exactly one-sixth of the circumference and one-sixth of the area — the same reason a regular hexagon fits perfectly inside a circle.

Radius 21 cm is a favourite CBSE choice because 21 is a multiple of 7, making $\pi = \frac{22}{7}$ cancel cleanly and giving whole-number answers.

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Frequently Asked Questions

What is the formula for the length of an arc?

Arc length = $(\theta/360°) \times 2\pi r$ , where θ is the central angle and r is the radius. It is the corresponding fraction of the full circumference 2πr.

Why do we use π = 22/7 here?

Because the radius 21 is a multiple of 7, using $\frac{22}{7}$ cancels neatly to give a whole-number answer. The question implicitly expects this standard value.

How is arc length different from sector area?

Arc length is a distance measured in cm along the boundary, using $2\pi r$ ; sector area is a region measured in cm² using $\pi r^2$ . Mixing them is a common error.