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

Areas Related to Circles

Length of an Arc of a Sector

Options

(A)22 cm

(B)44 cm

(D)11 cm

Official Answer

(a) 22 cm — Arc length = $\left(\frac{\theta}{360°}\right) \times 2\pi r = \left(\frac{60}{360}\right) \times 2 \times \frac{22}{7} \times 21 = 22$ cm.

arc lengthsector of a circlecentral anglecircumferenceπ = 22/7radius

Marking Scheme

11 mark: correctly selecting option (a) 22 cm using the formula $\left(\frac{\theta}{360°}\right) \times 2\pi r$ with $\theta = 60°, r = 21$ cm.
2No partial marks for MCQs; working may be shown for verification only.

Hint

Arc length = $\left(\frac{\theta}{360°}\right) \times 2\pi r$ , and choose $\pi = \frac{22}{7}$ since the radius 21 is a multiple of 7.

Quick Oral Answer

Arc length of a sector equals $\left(\frac{\theta}{360°}\right)$ times the full circumference $2\pi r$ ; substituting $\theta=60°$ and $r=21$ with $\pi=\frac{22}{7}$ gives 22 cm.

Analysis & Explanation

A direct application of the sector arc-length formula with π taken as 22/7.

Concept

Arc length of a sector = $\left(\frac{\theta}{360°}\right) \times 2\pi r$ , where θ is the central angle and r the radius.

Key points

Here $\theta = 60°$ , $r = 21$ cm, so Arc length = $\left(\frac{60}{360}\right) \times 2 \times \frac{22}{7} \times 21 = 22$ cm.
$r = 21$ is a multiple of 7, signalling that $\pi = \frac{22}{7}$ gives a clean answer.

Common mistakes

Confusing the arc-length formula with the sector-area formula (θ/360 × πr²), computing an area instead of a length.
Using πr instead of 2πr, forgetting the factor of 2 from the full circumference.

Real-world

Used to calculate lengths of curved boundaries — a curved road divider, the rim length of a pie-shaped wheel sector, or the arc traced by a rotating machine part.

Common Mistakes

1Using the sector area formula ( $\frac{\theta}{360} \times \pi r^2$ ) instead of the arc length formula ( $\frac{\theta}{360} \times 2\pi r$ ).
2Forgetting the factor of 2 in $2\pi r$ and using πr instead, halving the correct answer to give the distractor 11 cm.
3Using $\pi = 3.14$ instead of $\frac{22}{7}$ , leading to a non-clean decimal answer and possible rounding errors that don't match any given option cleanly.

Interesting Facts

The arc length formula $\left(\frac{\theta}{360}\right) \times 2\pi r$ comes directly from the fact that a full circle is a 360-degree sector, so any smaller angle is a fraction of the total circumference.

Ancient Babylonian astronomers divided the circle into 360 parts partly because 360 is close to the number of days in a year and has many divisors, a convention still used today for measuring arcs and sectors.

Using 21 as the radius in such problems is common in Indian textbooks because it pairs neatly with the approximation $\frac{22}{7}$ for $\pi$ , giving whole-number answers.

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Frequently Asked Questions

What is the formula for the length of an arc of a sector?

Arc length = $\left(\frac{\theta}{360°}\right) \times 2\pi r$ , where θ is the central angle in degrees and r is the radius.

Why is π taken as $\frac{22}{7}$ here and not 3.14?

Because the radius (21 cm) is a multiple of 7, using $\frac{22}{7}$ gives a clean, exact whole-number answer, which is the intended CBSE approach.