MathsICSE Class 8 · Chapter 19

Circle ICSE Class 8: Centre, Radius, Diameter, Chord, Arc, Sector and Segment

Updated 15 October 2026 · Fact-checked

A circle is the set of all points in a plane at the same distance, the radius, from a fixed point, the centre. ICSE Class 8 names its parts: diameter (twice the radius), chord, arc, sector (between two radii and an arc) and segment (between a chord and an arc), each with minor and major parts.

What Circle is about

A circle is the set of all points in a plane that are at the same distance from a fixed point. The fixed point is the centre ( $O$ ), and the fixed distance is the radius ( $r$ ).

The CISCE Class VIII curriculum asks you to know and use these terms:

centre, radius and diameter,
chord and arc,
sector and segment.

You should be able to draw and label each part, tell minor from major parts, decide whether a point is inside, on or outside a circle, and work out the angle or fraction of the circle that a sector takes up. These ideas are used again in the next chapter for the area of a semicircle, and later for pie charts.

A useful fact to keep in mind: the angle all the way round the centre is $360°$ , so a sector of angle $θ$ is $\frac{θ}{360}$ of the whole circle.

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Circle sits in the Geometry theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Centre, radius and diameter

The radius is a line segment from the centre to any point on the circle (and also its length, $r$ ). All radii of a circle are equal.
The diameter is a chord through the centre. Its length is $d = 2 r$ , and it is the longest chord.
A circle has infinitely many radii and infinitely many diameters.

A point $P$ is:

inside the circle if $O P < r$ ,
on the circle if $O P = r$ ,
outside the circle if $O P > r$ .

Circles with the same centre but different radii are concentric.

Worked example

A circle has radius 6.5 cm. Find its diameter. Points $P$ , $Q$ , $R$ are 6 cm, 6.5 cm and 7.2 cm from the centre. Where does each lie?

Try it first, then show the solutionSolution

Diameter $= 2 \times 6.5 = 13$ cm.
$P$ : $6 < 6.5$ , inside. $Q$ : $6.5 = 6.5$ , on the circle.
$R$ : $7.2 > 6.5$ , outside.

Answer: 13 cm; $P$ inside, $Q$ on, $R$ outside

Chord

A chord is a line segment joining any two points on the circle.

A diameter is a special chord that passes through the centre.
Every diameter is a chord, but most chords are not diameters.
The closer a chord is to the centre, the longer it is; the diameter is the longest.

Through $n$ points on a circle you can draw $\frac{n ( n - 1 )}{2}$ chords (each pair of points gives one).

Worked example

Six points are marked on a circle. How many chords can be drawn joining them in pairs?

Try it first, then show the solutionSolution

Each point can be joined to the 5 others: $6 \times 5 = 30$ joins.
Each chord has been counted twice (once from each end): $30 \div 2 = 15$ .

Answer: 15 chords

Arc: minor, major and semicircle

An arc is a part of the circle between two points on it. Two points $A$ and $B$ split the circle into two arcs:

the minor arc (the shorter one), written $A B ⌢$ ,
the major arc (the longer one), named with a third point on it, such as $A C B ⌢$ .

If $A$ and $B$ are ends of a diameter, both arcs are equal and each is a semicircle.

The angle that an arc makes at the centre tells its size: minor arc angle $+$ major arc angle $= 360°$ .

Worked example

Arc $A B$ makes an angle of $100°$ at the centre. What angle does the major arc make, and what fraction of the circle is the minor arc?

Try it first, then show the solutionSolution

Major arc angle $= 360° - 100° = 260°$ .
Minor arc fraction $= \frac{100}{360} = \frac{5}{18}$ .

Answer: $260°$ ; $\frac{5}{18}$

Sector

A sector is the region bounded by two radii and an arc (like a slice of pizza).

The angle between the two radii is the sector angle $θ$ .
A sector with angle $θ$ is $\frac{θ}{360}$ of the whole circle.
The minor sector has angle less than $180°$ ; the major sector is the rest ( $360° - θ$ ).
A sector of $90°$ is a quadrant (quarter circle); a sector of $180°$ is a semicircle.

Equal cuts from the centre make equal sectors: $n$ equal slices each have angle $\frac{360°}{n}$ .

Worked example

A sector has angle $40°$ . What fraction of the circle is it, and what is the angle of the major sector?

Try it first, then show the solutionSolution

Fraction $= \frac{40}{360} = \frac{1}{9}$ .
Major sector angle $= 360° - 40° = 320°$ .

Answer: $\frac{1}{9}$ ; $320°$

Segment

A segment is the region bounded by a chord and an arc.

A chord divides the circle into two segments: the minor segment (smaller) and the major segment (larger).
A diameter divides the circle into two equal segments, each a semicircle.

Do not mix up the two regions:

sector: two radii + arc (it touches the centre);
segment: chord + arc (the centre is not on its boundary, except for a semicircle).

A semicircle is the only region that is both a sector and a segment.

Worked example

A circle is cut along radii $O A$ and $O B$ and also along chord $A B$ , with $∠ A O B = 130°$ . Name the pieces.

Try it first, then show the solutionSolution

Triangle $O A B$ is cut off between the radii and the chord.
Between chord $A B$ and the minor arc lies the minor segment.
The rest of the circle (outside the radii) is the major sector with angle $360° - 130° = 230°$ .

Answer: Triangle $O A B$ , the minor segment, and the major sector ( $230°$ )

Drawing and labelling a circle

Draw-and-label checklist:
1. Mark the centre $O$ and draw the circle with compasses set to the radius.
2. Draw a radius $O A$ and a diameter $P Q$ through $O$ ; label them.
3. Draw a chord $C D$ not through $O$ .
4. Shade a sector (two radii and the arc between) and a segment (a chord and its arc) in different ways, and label minor and major parts.
5. Write the radius length next to the radius.

Keep the compass width fixed while drawing, or the curve will not close.

Worked example

In 35 minutes, through what angle does the minute hand of a clock turn? What part of the clock face does it sweep?

Try it first, then show the solutionSolution

60 minutes $= 360°$ , so 1 minute $= 6°$ .
35 minutes $= 210°$ , a sector of $\frac{210}{360} = \frac{7}{12}$ of the face (a major sector, as $210° > 180°$ ).

Answer: $210°$ ; $\frac{7}{12}$ of the face

Maths in real life: Circle

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Radius and diameter

Suppose each spoke of a bicycle wheel, from the hub to the rim, is 30 cm long.

What is the diameter of the wheel?

Try it first, then show the solutionSolution

The spoke is a radius: $r = 30$ cm.
$d = 2 r = 60$ cm.

Answer: 60 cm

Memory hook: Spokes are radii.

Equal sectors

Imagine cutting a round cake from the centre into 6 equal pieces.

What is the angle of each piece?

Try it first, then show the solutionSolution

A full turn is $360°$ .
Each piece $= 360° \div 6 = 60°$ .

Answer: $60°$

Memory hook: Equal slices share 360° equally.

Inside, on or outside

Suppose a goat is tied to a pole with a 4 m rope. A ball lies 3.5 m from the pole and a bucket 4.5 m from it.

Which can the goat reach?

Try it first, then show the solutionSolution

The goat can reach everything inside or on a circle of radius 4 m.
Ball: $3.5 < 4$ , inside, reachable. Bucket: $4.5 > 4$ , outside, not reachable.

Answer: The ball but not the bucket

Memory hook: Compare the distance with the radius.

Topics in this chapter

Chord, Arc, Sector and Segment Problems

Circle formula sheet

Diameter: $d = 2 r$; Longest chord.
Angle round the centre: $minor + major = 360°$; For arcs and sectors.
Fraction of a circle: $\frac{θ}{360°}$; Sector or arc of angle θ.
Equal sectors: $θ = \frac{360°}{n}$; n equal slices.
Clock hands: $minute hand: 6° per minute$; Hour hand: 30° per hour, 0.5° per minute.

How the school exam asks Circle

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Definitions	Define a chord, an arc, a sector and a segment.	Name the boundary of each: chord = two points; sector = two radii + arc; segment = chord + arc.
Radius and diameter	Find the diameter from the radius, or decide if a point is inside, on or outside.	d = 2r; compare the distance with r.
Sector angles	A sector is 72°: what fraction is it; what is the major sector?	θ/360 and 360° − θ.
Diagram labelling	Draw a circle and show a radius, a chord, a minor segment and a major sector.	Follow the draw-and-label checklist.
Applications	Clock hands, pizza slices, Ferris wheels.	Use 360° for a full turn and divide equally.

Tricks to remember Circle

Short memory aids. Say each one aloud once, then use it on a question the same day.

Sector = slice, segment = chopped edge
A sector is a pizza slice (from the centre). A segment is what you get by chopping off the edge with one straight cut (a chord).
Use it when: Never mixing the two.
Diameter is double
d = 2r: the diameter goes across, through the centre, two radii long.
Use it when: Radius–diameter questions.
Minor + major = 360
The two arcs (or sectors) made by the same two radii always add up to a full turn.
Use it when: Major arc and major sector angles.
Every diameter is a chord
But not every chord is a diameter: only the ones through the centre.
Use it when: True/false and assertion questions.

Circle: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Longest chord · 1 mark · MCQExam level

The longest chord of a circle is:
(a) a radius
(b) a diameter
(c) an arc
(d) a tangent

Try it first, then show the solutionSolution

A radius has only one end on the circle, and an arc is curved, so neither is a chord.
The chord through the centre, the diameter, is the longest.

Answer: (b)

Example 2: Inside, on or outside · 3 marks · Short answerExam level

A circle has radius 4.5 cm. Find its diameter, and say where points 5 cm, 4.5 cm and 3 cm from the centre lie.

Try it first, then show the solutionSolution

$d = 2 \times 4.5 = 9$ cm.
5 cm: outside; 4.5 cm: on the circle.
3 cm: inside.

Answer: 9 cm; outside, on, inside

Example 3: A sector as a fraction · 3 marks · Short answerExam level

A sector of a circle has angle $72°$ . What fraction of the circle is it, and what is the angle of the major sector?

Try it first, then show the solutionSolution

Fraction $= \frac{72}{360}$ .
$= \frac{1}{5}$ .
Major sector $= 360° - 72° = 288°$ .

Answer: $\frac{1}{5}$ ; $288°$

Example 4: Minute hand · 3 marks · Short answerHarder exam level

Through what angle does the minute hand of a clock turn in 20 minutes? What fraction of the clock face does it sweep, and is this a minor or a major sector?

Try it first, then show the solutionSolution

1 minute $= 360° \div 60 = 6°$ ; 20 minutes $= 120°$ .
Fraction $= \frac{120}{360} = \frac{1}{3}$ .
$120° < 180°$ , so it is a minor sector.

Answer: $120°$ ; $\frac{1}{3}$ ; minor sector

Example 5: Concentric circles · 4 marks · Long answerHarder exam level

Two concentric circles have radii 3 cm and 5 cm. (i) Find both diameters. (ii) How wide is the ring between them? (iii) Where does a point 4 cm from the centre lie?

Try it first, then show the solutionSolution

(i) Diameters: 6 cm and 10 cm.
(ii) Ring width $= 5 - 3 = 2$ cm.
(iii) $4 > 3$ , so it is outside the smaller circle.
$4 < 5$ , so it is inside the larger circle: it lies in the ring.

Answer: (i) 6 cm, 10 cm (ii) 2 cm (iii) in the ring

Example 6: Pizza slices · 4 marks · Long answerHarder exam level

A round pizza is cut from the centre into 8 equal slices. (i) What is the angle of each slice? (ii) Riya takes 3 slices next to each other. What angle does her share make, and what angle is left?

Try it first, then show the solutionSolution

(i) $360° \div 8 = 45°$ .
(ii) 3 slices $= 3 \times 45° = 135°$ (a minor sector).
Left $= 360° - 135°$ .
$= 225°$ (a major sector).

Answer: (i) $45°$ (ii) $135°$ ; $225°$ left

Example 7: Sectors and segments together · 5 marks · Long answerHOTS / competency

In a circle with centre $O$ , $∠ A O B = 100°$ . (i) Find the angle of the major arc. (ii) What fraction of the circle is the minor sector $A O B$ ? (iii) The circle is cut along $O A$ , $O B$ and chord $A B$ . Name the three pieces. (iv) Which region is both a sector and a segment?

Try it first, then show the solutionSolution

(i) $360° - 100° = 260°$ .
(ii) $\frac{100}{360} = \frac{5}{18}$ .
(iii) Triangle $O A B$ ; the minor segment between chord $A B$ and the minor arc; the major sector of angle $260°$ .
(iv) A semicircle: it is bounded by two radii (forming a diameter) and an arc, and also by a chord (the diameter) and an arc.
So only when $∠ A O B = 180°$ do sector and segment coincide.

Answer: (i) $260°$ (ii) $\frac{5}{18}$ (iii) triangle, minor segment, major sector (iv) a semicircle

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

The Ferris wheel

Suppose a Ferris wheel has radius 10 m and 12 cabins fixed at equal spaces round its rim. It makes one full turn every 10 minutes.

(i) · 3 marks
What is the diameter of the wheel? Is a straight line joining two opposite cabins a radius, a chord or a diameter?
Answer
Answer: 20 m; a diameter (also a chord)
$d = 2 \times 10 = 20$ m. Opposite cabins are joined through the centre, so the line is a diameter, which is a chord.
(ii) · 3 marks
What angle at the centre separates neighbouring cabins? What angle does the sector from cabin 1 to cabin 3 make?
Answer
Answer: $30°$ ; $60°$
$360° \div 12 = 30°$ . Cabin 1 to cabin 3 is two gaps: $60°$ .
(iii) · 4 marks
Through what angle does the wheel turn in 2.5 minutes? How many cabin positions does each cabin move?
Answer
Answer: $90°$ ; 3 positions
In 10 minutes it turns $360°$ , so in 2.5 minutes it turns $\frac{2.5}{10} \times 360° = 90°$ . $90° \div 30° = 3$ positions.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

A circle has radius 5 cm. Points $A$ , $B$ , $C$ are 4.9 cm, 5 cm and 5.2 cm from the centre. State where each lies, and find the diameter. (3 marks)

Step	Marks
$A$ : $4.9 < 5$ , inside	1
$B$ : on the circle; $C$ : $5.2 > 5$ , outside	1
Diameter $= 10$ cm	1

A circle is divided into three sectors whose angles are in the ratio $2 : 3 : 4$ . Find each angle and say which, if any, is a major sector. (4 marks)

Step	Marks
$2 x + 3 x + 4 x = 360°$	1
$x = 40°$	1
Angles $80°, 120°, 160°$	1
All are less than $180°$ , so none is a major sector	1

At 3:00 a clock's hour hand points to 3. Find the angle between the hands at 3:20. (5 marks)

Step	Marks
Minute hand moves $6°$ per minute: at 3:20 it is $120°$ from 12	1
Hour hand moves $30°$ per hour, i.e. $0.5°$ per minute	1
At 3:00 the hour hand is at $90°$ ; in 20 minutes it moves $10°$	1
At 3:20 the hour hand is at $100°$	1
Angle between the hands $= 120° - 100° = 20°$	1

Where students lose marks in Circle

The slips examiners see most often, and what to write instead.

Mistake: Calling a radius a chord
A chord has both ends on the circle; a radius has one end at the centre.
Do this: Check where the two ends of the segment are.
Usually costs: 1 mark.
Mistake: Mixing up sector and segment
A sector is cut by two radii; a segment by one chord.
Do this: Sector = slice from the centre; segment = chopped-off edge.
Usually costs: 1 mark per label.
Mistake: Forgetting the major part
Two points give two arcs and two sectors.
Wrong working
Arc $A B = 100°$ , so the circle is $100°$
Right working
Minor arc $100°$ , major arc $260°$
Do this: Write both, adding to $360°$ .
Usually costs: 1 mark.
Mistake: Halving instead of doubling
$d = 2 r$ , so $r = \frac{d}{2}$ .
Wrong working
$r = 7$ cm, so $d = 3.5$ cm
Right working
$d = 14$ cm
Do this: The diameter is always the bigger number.
Usually costs: 1 mark.
Mistake: Unlabelled diagrams
Marks are for showing and naming each part.
Do this: Label the centre, each line and each shaded region.
Usually costs: Up to 2 marks.

Practice questions on Circle

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQExam level
The diameter of a circle with radius 7 cm is:
(a) 3.5 cm
(b) 7 cm
(c) 14 cm
(d) 21 cm
Hint
$d = 2 r$ .
Answer
(c)
Q2 · 1 mark · Very short answerExam level
The diameter of a circle is 15 cm. Find its radius.
Hint
Halve it.
Answer
7.5 cm.
Q3 · 2 marks · Short answerExam level
Define a chord and an arc of a circle.
Hint
Straight versus curved.
Answer
A chord is a line segment joining any two points on a circle. An arc is a part of the circle between two points on it.
Q4 · 3 marks · Short answerHarder exam level
A circle is divided into three sectors of $90°$ , $120°$ and the rest. Find the third angle and the fraction of the circle each sector is.
Hint
Angles add to 360°.
Answer
Third $= 150°$ . Fractions $\frac{1}{4}$ , $\frac{1}{3}$ , $\frac{5}{12}$ .
Q5 · 1 mark · Assertion-reasonHarder exam level
Assertion (A): Every diameter of a circle is a chord.
Reason (R): A chord joins any two points on the circle.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Where are the ends of a diameter?
Answer
(c)
Q6 · 1 mark · Very short answerHarder exam level
How many diameters can a circle have? How many radii make up one diameter?
Hint
Any line through the centre.
Answer
Infinitely many diameters; each is made of 2 radii.
Q7 · 2 marks · Short answerHarder exam level
Through what angle does the minute hand turn in 45 minutes, and what fraction of the face does it sweep?
Hint
$6°$ per minute.
Answer
$270°$ ; $\frac{3}{4}$ of the face (a major sector).
Q8 · 2 marks · Short answerHarder exam level
A circle has radius 3 cm. Points are 2.9 cm, 3 cm and 3.1 cm from the centre. State where each lies.
Hint
Compare with 3.
Answer
Inside; on the circle; outside.
Q9 · 4 marks · Long answerHarder exam level
A circle is cut into three sectors with angles in the ratio $1 : 2 : 3$ . Find the angles. Which sector is also a segment, and why?
Hint
Total 360°.
Answer
$6 x = 360°$ , $x = 60°$ : angles $60°, 120°, 180°$ . The $180°$ sector is a semicircle; its two radii form a diameter (a chord), so it is also a segment.
Q10 · 1 mark · Very short answerHarder exam level
Name the region that is both a sector and a segment of a circle.
Hint
Cut along a diameter.
Answer
A semicircle.
Q11 · 3 marks · Short answerHOTS / competency
In 2 hours, through what angle does (i) the hour hand (ii) the minute hand of a clock turn?
Hint
Hour hand: 30° per hour.
Answer
(i) $2 \times 30° = 60°$ (ii) 2 full turns $= 720°$ .
Q12 · 3 marks · Short answerHOTS / competency
3 points on a circle can be joined by 3 chords. How many chords join 5 points? Explain your method.
Hint
Each pair gives one chord.
Answer
Each of 5 points joins 4 others: $5 \times 4 = 20$ , but each chord is counted twice, so $20 \div 2 = 10$ chords.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is the radius of a circle?Show answer
The distance from the centre to any point on the circle.
2.If r = 6 cm, what is d?Show answer
12 cm.
3.What is a chord?Show answer
A line segment joining two points on the circle.
4.What is a sector?Show answer
The region between two radii and an arc.
5.What is a segment?Show answer
The region between a chord and an arc.
6.What is the angle of a semicircle at the centre?Show answer
180°.
7.Minor arc 70°: major arc?Show answer
290°.
8.How many diameters does a circle have?Show answer
Infinitely many.

Best order to study it

Learn centre, radius and diameter, and d = 2r.
Learn chord and arc, and minor and major arcs.
Learn sector and segment and how they differ.
Practise sector angles as fractions of 360°.
Finish with diagrams and applications such as clocks and wheels.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Can you define each part in one line?
Did you use d = 2r correctly?
Did you compare distance with the radius for inside/on/outside?
Do minor and major parts add to 360°?
Did you label sectors and segments correctly?
Is your diagram drawn with compasses and labelled?

Circle: frequently asked questions

Is the diameter a chord?

Yes. It joins two points on the circle, and it is the longest chord because it passes through the centre.

What is the difference between an arc and a chord?

An arc is a curved part of the circle itself. A chord is the straight line joining the two ends of that arc.

Can a sector be bigger than a semicircle?

Yes. A major sector has an angle greater than 180°.

What is a quadrant?

A sector of angle 90°, one quarter of the circle.

Do I need the formula for the area of a circle in this chapter?

This chapter is about the parts of a circle. Areas of semicircles are used in the next chapter, Area of Plane Figures.

Based on: CISCE Curriculum for Upper Primary Classes (VI-VIII), Mathematics (Class VIII); CISCE RDCD: Curriculum Pre-School to Class VIII