Class 10 · Chapter 11 · Mensuration unit (10 of 80 marks)

Areas Related to Circles Class 10: notes and important questions

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 37 board-style questions.

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37 questions
10 multiple choice, 3 assertion–reason, 8 very short answer, 8 short answer, 5 long answer, 3 case study
About 9 hours to master

Mensuration unit: 10 of 80 theory marks (Areas Related to Circles and Surface Areas and Volumes).

NCERT textbook: Chapter 11 on ncert.nic.in (PDF)

Revision notes

Basic results (circle of radius $r$ )

Circumference $= 2 π r$ ; area $= π r^{2}$ .
For a sector with central angle $θ$ (in degrees):
- Length of arc $l = \frac{θ}{360^{\circ}} \times 2 π r$
- Area of sector $= \frac{θ}{360^{\circ}} \times π r^{2} = \frac{1}{2} l r$
- Perimeter of sector $= 2 r + l$
Major sector area $= π r^{2} -$ minor sector area (its angle is $360^{\circ} - θ$ ).
Segment (the region between a chord and its arc): minor segment area $=$ area of sector $-$ area of $△ O A B$ .
- $θ = 60^{\circ}$ : triangle is equilateral, area $= \frac{\sqrt{3}}{4} r^{2}$
- $θ = 90^{\circ}$ : area $= \frac{1}{2} r^{2}$
- $θ = 120^{\circ}$ : area $= \frac{\sqrt{3}}{4} r^{2}$ (height $\frac{r}{2}$ , base $r \sqrt{3}$ )
Major segment area $= π r^{2} -$ minor segment area.

Worked example 1

Find the area of a sector of angle $40^{\circ}$ in a circle of radius 21 cm $(π = \frac{22}{7})$ .

$\frac{40}{360} \times \frac{22}{7} \times 441 = \frac{1}{9} \times 1386 = 154 {cm}^{2}$ .

Worked example 2

A chord of a circle of radius 10 cm subtends $60^{\circ}$ at the centre. Find the minor segment area ( $π = 3.14, \sqrt{3} = 1.73$ ).

Sector $= \frac{1}{6} \times 3.14 \times 100 = 52.33$ ; triangle $= \frac{1.73}{4} \times 100 = 43.25$ ; segment $\approx 9.08 {cm}^{2}$ .

Worked example 3

The tip of a 12 cm minute hand moves for 25 minutes. How far does it travel? Angle $= 25 \times 6^{\circ} = 150^{\circ}$ ; arc $= \frac{150}{360} \times 2 \times 3.14 \times 12 = 31.4$ cm.

Handy facts

The minute hand turns $6^{\circ}$ per minute; the hour hand turns ${0.5}^{\circ}$ per minute.
Sector areas of the same circle are in the ratio of their angles. For a fixed angle, sector area is proportional to $r^{2}$ and arc length to $r$ .

Common errors

Adding the triangle to the sector instead of subtracting it to get the minor segment.
Using the diameter in place of the radius.
Forgetting the two radii when finding the perimeter of a sector.
Using $π = 3.14$ when the question says $\frac{22}{7}$ , or rounding too early.

Board-exam tips

Syllabus limit: segment problems use central angles of 60°, 90° and 120° only.
Write the formula first, then substitute. This earns the method mark even if the arithmetic slips.
If the answer is in terms of $π$ and $\sqrt{3}$ , leave it exact unless a value is given.

Topics in this chapter: Area of a sector · Length of an arc · Area of a segment.

Formulas for this chapter: Chapter 11 on the Class 10 formula sheet (free PDF and A3 poster).

Areas Related to Circles Class 10: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 190 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceArea of a sector

The area of a sector of angle $60^{\circ}$ in a circle of radius $6$ cm is

Show worked answer

$\frac{60}{360} \times π \times 6^{2} = 6 π {cm}^{2}$ A1
Correct option: (c) $6 π {cm}^{2}$

Q2·1 mark·Multiple choiceLength of an arc

The length of the arc that subtends $90^{\circ}$ at the centre of a circle of radius $14$ cm is (take $π = \frac{22}{7}$ )

Q3·1 mark·Assertion–reasonArea of a sector

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): A circle whose circumference is $44$ cm has area $154 {cm}^{2}$ (take $π = \frac{22}{7}$ ).

Reason (R): The area of a sector of angle $θ$ in a circle of radius $r$ is $\frac{θ}{360^{\circ}} \times π r^{2}$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerArea of a sector

Find the area of a sector of angle $120^{\circ}$ in a circle of radius $21$ cm. (Take $π = \frac{22}{7}$ .)

Show worked answer

Answer:

462 {cm}^{2}

Area $= \frac{120}{360} \times \frac{22}{7} \times 21 \times 21$ M1
$= \frac{1}{3} \times 1386 = 462 {cm}^{2}$ A1

Q5·2 marks·Short answerLength of an arc

The bob of a pendulum swings through an angle of $60^{\circ}$ and moves along an arc $11$ cm long. Find the length of the pendulum. (Take $π = \frac{22}{7}$ .)

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerArea of a segment

A chord of a circle of radius $6$ cm subtends an angle of $60^{\circ}$ at the centre. Find the area of the minor segment, leaving $π$ and surds in your answer.

Show worked answer

Answer:

(6 π - 9 \sqrt{3}) {cm}^{2}

Area of the sector $= \frac{60}{360} \times π \times 36 = 6 π {cm}^{2}$ A1
The triangle formed by the chord and the radii is equilateral with side $6$ cm: area $= \frac{\sqrt{3}}{4} \times 36 = 9 \sqrt{3} {cm}^{2}$ M1
Minor segment $= (6 π - 9 \sqrt{3}) {cm}^{2}$ A1

Q7·3 marks·Short answerArea of a sector

A goat is tied to a peg at one corner of a rectangular field $30$ m by $25$ m with a rope $14$ m long. Find the area of the field the goat can graze. How much more area could it graze if the rope were $21$ m long? (Take $π = \frac{22}{7}$ .)

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerArea of a sector

A car's wiper arm is $35$ cm long. Its rubber blade covers the part of the arm from $14$ cm to $35$ cm from the pivot, and the arm turns through $120^{\circ}$ . (a) Find the area of glass cleaned in one sweep. (b) Find the length of the arc traced by the tip of the blade. (Take $π = \frac{22}{7}$ .)

Show worked answer

Answer:

1078 {cm}^{2}

;

\frac{220}{3}

(a) The cleaned region is the sector of radius $35$ cm minus the sector of radius $14$ cm, both of angle $120^{\circ}$ M1
Larger sector $= \frac{1}{3} \times \frac{22}{7} \times 1225 = \frac{3850}{3} {cm}^{2}$ A1
Smaller sector $= \frac{1}{3} \times \frac{22}{7} \times 196 = \frac{616}{3} {cm}^{2}$ A1
Area cleaned $\begin{aligned} = \frac{3850 - 616}{3} & = \frac{3234}{3} \\ = 1078 {cm}^{2} \end{aligned}$ A1
(b) Arc $= \frac{1}{3} \times 2 \times \frac{22}{7} \times 35 = \frac{220}{3} \approx 73.3$ cm A1

Q9·5 marks·Long answerArea of a segment

The glass at the top of an arched window is the minor segment cut from a circle of radius $60$ cm by a chord $A B$ that subtends $120^{\circ}$ at the centre $O$ . Find (a) the length of the arc $A B$ , (b) the area of the sector $O A B$ , (c) the area of the glass, and (d) the length of edging needed for the arc and the chord together. Leave $π$ and surds in your answers.

Section E style · Case study (4 marks)

Q10·4 marks·Case studyArea of a sector

Fan-shaped stage. An open-air stage is shaped as a sector of a circle of radius $21$ m with an angle of $60^{\circ}$ at the centre. (Take $π = \frac{22}{7}$ .)

Find the length of the curved front edge of the stage. [1 mark]
Find the floor area of the stage. [1 mark]
The stage is widened so that its angle becomes $90^{\circ}$ , with the same radius. How much floor area is added? [2 marks]

Show worked answer

$\frac{60}{360} \times 2 \times \frac{22}{7} \times 21 = 22$ m A1
$\frac{60}{360} \times \frac{22}{7} \times 21^{2} = 231 m^{2}$ A1
The added part is a sector of angle $30^{\circ}$ : $\frac{30}{360} \times \frac{22}{7} \times 441$ M1
$= 115.5 m^{2}$ A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 10 assertion–reason questions · the Class 10 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can find the length of an arc and the area of a sector for any angle.

Read first: Basic results (circle of radius r); Worked example 1 11 practice questions · checkpoint: 3 questions, 5 marks, pass 80%

Practise step 1

Step 2

Board standard

You can find segment areas for 60°, 90° and 120°, and solve clock-hand and sector-perimeter questions.

Read first: Basic results; Handy facts; Worked examples 2-3 14 practice questions · checkpoint: 3 questions, 7 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write complete long answers on shaded regions, combining sectors, triangles and squares without missing a piece.

Read first: Worked examples 1-3; Board-exam tips 7 practice questions · checkpoint: 3 questions, 13 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can work backwards from a given area or perimeter to the radius or angle, and handle major segments.

Read first: Handy facts; Common errors 5 practice questions · checkpoint: 3 questions, 11 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 8 of the 37 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceLength of an arc

The length of an arc of a circle of radius 21 cm subtending an angle of $60^{\circ}$ at the centre is $(π = \frac{22}{7})$

(a)44 cm
(b)22 cm
(c)231 cm
(d)66 cm

Q2·1 mark·Multiple choiceLength of an arc

The perimeter of a sector of radius $r$ and central angle $θ$ is

(a) $\frac{θ}{360} \times 2 π r$
(b) $2 r + \frac{θ}{360} \times 2 π r$
(c) $r + \frac{θ}{360} \times 2 π r$
(d) $2 r + \frac{θ}{360} \times π r^{2}$

Q3·2 marks·Very short answerLength of an arc

Find the length of the arc of a circle of radius 4.2 cm that subtends an angle of $150^{\circ}$ at the centre. $(π = \frac{22}{7})$

Worked solution·2 marks·Very short answerArea of a segment

A chord of a circle of radius 12 cm subtends an angle of $60^{\circ}$ at the centre. Find the area of the minor segment in terms of $π$ and $\sqrt{3}$ .

Full worked solution Free, no sign-in

Sector $= \frac{1}{6} π (144) = 24 π$ ; triangle (equilateral) $= \frac{\sqrt{3}}{4} (144) = 36 \sqrt{3}$ M1

Segment $= (24 π - 36 \sqrt{3}) {cm}^{2}$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original geometry problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I102 · Problem I20 · Problem I88 · Problem I95. All 28 →

Themed lesson: Women in maths maths lesson · Pi Day maths lesson · Pi Approximation Day maths lesson (starter, activity and worked answers). All themed maths

Knowledge organiser: Areas Related to Circles on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: Arc length · Area of a circle · Area of a sector. All formulas A–Z

Where marks are lost in Areas Related to Circles

Adding the triangle to the sector for a minor segment. Fix: minor segment = sector − triangle; write the word equation first.
Perimeter of a sector without the two radii. Fix: perimeter = arc + 2r.
Using the diameter as the radius. Fix: underline 'diameter' in the question and halve it on the first line.
Using π = 3.14 when the question says 22/7 (or the reverse), or rounding √3 early. Fix: circle the values given in the question and use exactly those.
Shaded-region questions with a missing piece. Fix: list the pieces in words (square − 4 quadrants, and so on) before calculating.
Missing units, especially cm² for area and cm for arc length. Fix: write the unit on every final answer.

Exam strategy: how answers are step-marked · competency-based questions

Areas Related to Circles in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 12, 13, 31, 38
Sample paper 2 (Board standard): questions 14, 25
Sample paper 3 (Above board): questions 16, 24
Sample paper 4 (Above board): questions 14, 20
Sample paper 5 (95+ challenge): questions 17, 24

Want it against the clock? Take a timed 30-mark chapter test on Areas Related to Circles, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Areas Related to Circles: common questions

What is in CBSE Class 10 Maths Chapter 11, Areas Related to Circles?

The topics are: Area of a sector, Length of an arc, Area of a segment. This page has revision notes for each, and 37 board-style practice questions with step-marked solutions.

How many marks is Areas Related to Circles worth?

It is in the Mensuration unit, which carries 10 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Areas Related to Circles?

Adding the triangle to the sector for a minor segment. Fix: minor segment = sector − triangle; write the word equation first. Perimeter of a sector without the two radii. Fix: perimeter = arc + 2r.

Are the Areas Related to Circles important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

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Revision notes

Basic results (circle of radius r)

Worked example 1

Worked example 2

Worked example 3

Handy facts

Common errors

Board-exam tips

Areas Related to Circles Class 10: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Areas Related to Circles

Areas Related to Circles in our sample papers

Areas Related to Circles: common questions

Work through the Route to 95 with your progress saved

Basic results (circle of radius $r$ )