Question 141 markMCQSection A

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
(C)88 cm
(D)11 cm

Official answer

(a) 22 cm — Arc length = (θ360°)×2πr=(60360)×2×227×21=22\left(\frac{\theta}{360°}\right) \times 2\pi r = \left(\frac{60}{360}\right) \times 2 \times \frac{22}{7} \times 21 = 22 cm.

Answer keywords:arc lengthsector of a circlecentral anglecircumferenceπ = 22/7radius

Marking scheme

  1. 1.1 mark: correctly selecting option (a) 22 cm using the formula (θ360°)×2πr\left(\frac{\theta}{360°}\right) \times 2\pi r with θ=60°,r=21\theta = 60°, r = 21 cm.
  2. 2.No partial marks for MCQs; working may be shown for verification only.

Hint

Arc length = (θ360°)×2πr\left(\frac{\theta}{360°}\right) \times 2\pi r, and choose π=227\pi = \frac{22}{7} since the radius 21 is a multiple of 7.

Quick oral answer

Arc length of a sector equals (θ360°)\left(\frac{\theta}{360°}\right) times the full circumference 2πr2\pi r; substituting θ=60°\theta=60° and r=21r=21 with π=227\pi=\frac{22}{7} gives 22 cm.

Common mistakes

  1. 1.Using the sector area formula (θ360×πr2\frac{\theta}{360} \times \pi r^2) instead of the arc length formula (θ360×2πr\frac{\theta}{360} \times 2\pi r).
  2. 2.Forgetting the factor of 2 in 2πr2\pi r and using πr instead, halving the correct answer to give the distractor 11 cm.
  3. 3.Using π=3.14\pi = 3.14 instead of 227\frac{22}{7}, leading to a non-clean decimal answer and possible rounding errors that don't match any given option cleanly.

Analysis and explanation

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


Concept

  • Arc length of a sector = (θ360°)×2πr\left(\frac{\theta}{360°}\right) \times 2\pi r, where θ is the central angle and r the radius.

Key points

  • Here θ=60°\theta = 60°, r=21r = 21 cm, so Arc length = (60360)×2×227×21=22\left(\frac{60}{360}\right) \times 2 \times \frac{22}{7} \times 21 = 22 cm.
  • r=21r = 21 is a multiple of 7, signalling that π=227\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.

Interesting facts

The arc length formula (θ360)×2πr\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 227\frac{22}{7} for π\pi, giving whole-number answers.

Question details

Q14 of 56 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Standard)
Year
2026
Section
Section A
Marks
1
Type
MCQ
Chapter
Areas Related to Circles
Topic
Length of an Arc of a Sector
Bloom's level
apply
Syllabus topic
Unit VI: Geometry / Mensuration — Areas Related to Circles: Length of arc of a sector
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

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

Arc length = (θ360°)×2πr\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 227\frac{22}{7} here and not 3.14?

Because the radius (21 cm) is a multiple of 7, using 227\frac{22}{7} gives a clean, exact whole-number answer, which is the intended CBSE approach.