MathsICSE Class 8 · Circle

Chord, Arc, Sector and Segment: Problems on the Parts of a Circle (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

The longest chord of a circle is its diameter. A perpendicular from the centre to a chord bisects the chord, so $r^2 = d^2 + \left(\frac{c}{2}\right)^2$ for chord $c$ at distance $d$ from the centre. The circumference is $2\pi r$, so a semicircle has perimeter $\pi r + 2r$ and a sector of angle $\theta$ has arc $\frac{\theta}{360} \times 2\pi r$.

Understand Chord, Arc, Sector and Segment Problems

The CISCE curriculum covers the circle, centre, radius/diameter, arc, chord, sector and segment.

Parts.

Radius $r$ : centre to a point on the circle. Diameter $d = 2 r$ : a chord through the centre.
Chord: a line segment joining two points on the circle. The diameter is the longest chord.
Arc: a part of the circle. The smaller is the minor arc, the larger the major arc; if equal, each is a semicircle.
Sector: the region between two radii and the arc they cut off.
Segment: the region between a chord and its arc.

Circumference. $C = 2 π r = π d$ , with $π \approx \frac{22}{7}$ for multiples of $7$ .

Perimeters. Semicircle: $π r + 2 r$ (arc plus diameter). Sector with central angle $θ$ : arc $+ 2 r$ , where arc $= \frac{θ}{360} \times 2 π r$ (an extension used for practice, the same fraction idea as for a quarter-circle or semicircle).

Chord and centre. If a perpendicular from the centre meets a chord at its midpoint, then a right-angled triangle is formed with hypotenuse $r$ and legs $d$ (distance of the chord from the centre) and $\frac{c}{2}$ :
$ $r^{2} = d^{2} + (\frac{c}{2})^{2} .$ $
A longer chord is closer to the centre; the diameter ( $d = 0$ ) is the longest.

Two parallel chords. On the same side of the centre the distance between them is $∣ d_{1} - d_{2} ∣$ ; on opposite sides it is $d_{1} + d_{2}$ .

Chord, Arc, Sector and Segment Problems: formulas to remember

Circumference: $C = 2 π r$; Use $\frac{22}{7}$ when $r$ is a multiple of 7.
Chord and centre: $r^{2} = d^{2} + (\frac{c}{2})^{2}$; $d$ = distance from centre to chord.
Semicircle perimeter: $π r + 2 r$; Arc plus the diameter.

How to solve Chord, Arc, Sector and Segment Problems questions

Identify the part, draw the radius to the chord end, and use Pythagoras or the circumference.

1Draw and label the circle, centre and the part asked.
2For a chord, draw the perpendicular from the centre to the chord.
3Form a right-angled triangle and apply Pythagoras.
4For arcs and perimeters, use $2 π r$ and the correct fraction.
5State the answer with units.

Chord, Arc, Sector and Segment Problems in real life

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.

Circular garden path

Suppose a circular pond has radius 5 m and a straight plank 8 m long is laid across it with both ends on the edge.

How far is the plank from the centre of the pond?

Try it first, then show the solutionSolution

Half the plank $= 4$ m, radius $= 5$ m.
Distance $= 5^{2} - 4^{2} = 3$ m.

Answer: 3 m.

Memory hook: Radius is the hypotenuse; half the chord and the distance are the legs.

Tricks to remember Chord, Arc, Sector and Segment Problems

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

3-4-5 and 5-12-13
Many chord problems use Pythagorean triples: radius 5 with chord 8 gives distance 3; radius 13 with chord 24 gives distance 5.
Use it when: Quick checks.
Diameter is longest
No chord can be longer than the diameter.
Use it when: Comparing chords.

Chord, Arc, Sector and Segment Problems: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Longest chord · 1 mark · MCQBoard level

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

Try it first, then show the solutionSolution

A diameter passes through the centre.

Answer: (b)

Example 2: Semicircle perimeter · 3 marks · Short answerBoard level

A circle has radius 7 cm. Find its diameter, its circumference and the perimeter of a semicircle cut from it.

Try it first, then show the solutionSolution

Diameter $= 14$ cm.
Circumference $= 2 \times \frac{22}{7} \times 7 = 44$ cm.
Semicircle perimeter $= 22 + 14 = 36$ cm.

Answer: 14 cm; 44 cm; 36 cm

Example 3: Chord distance · 3 marks · Short answerHarder board level

A chord 8 cm long is drawn in a circle of radius 5 cm. Find its distance from the centre.

Try it first, then show the solutionSolution

The perpendicular from the centre bisects the chord: half $= 4$ cm.
$d = 5^{2} - 4^{2} = 9 = 3$ cm.

Answer: 3 cm

Example 4: Quarter-circle sector · 4 marks · Short answerHarder board level

A sector of a circle of radius 14 cm has a central angle of $9 0^{\circ}$ . Find the fraction of the circle it is, its arc length and its perimeter.

Try it first, then show the solutionSolution

Fraction $= \frac{90}{360} = \frac{1}{4}$ .
Circumference $= 2 \times \frac{22}{7} \times 14 = 88$ cm; arc $= \frac{88}{4} = 22$ cm.
Perimeter $= 22 + 14 + 14 = 50$ cm.

Answer: 1/4; 22 cm; 50 cm

Example 5: Find the radius from a chord · 4 marks · Long answerHOTS / competency

A chord 24 cm long is 5 cm from the centre of a circle. Find the radius and the diameter, and explain why the chord is not the longest.

Try it first, then show the solutionSolution

Half chord $= 12$ , distance $= 5$ .
$r = 1 2^{2} + 5^{2} = 169 = 13$ cm; diameter $= 26$ cm.
The chord (24 cm) is shorter than the diameter (26 cm), the longest chord.

Answer: r = 13 cm; diameter = 26 cm

Where students lose marks in Chord, Arc, Sector and Segment Problems

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

Mistake: Using the whole chord in the right-angled triangle
Only half the chord forms the leg.
Wrong working
$d = 5^{2} - 8^{2}$ (not defined).
Right working
$d = 5^{2} - 4^{2} = 3$ .
Do this: Mark the midpoint of the chord before writing.
Usually costs: Answer mark.
Mistake: Forgetting the diameter in the semicircle perimeter
The perimeter includes the straight edge.
Wrong working
Perimeter of semicircle with $r = 7$ : $22$ cm.
Right working
$22 + 14 = 36$ cm.
Do this: Add the straight edge.
Usually costs: Answer mark.

Exam tips

Draw the radius to the end of the chord to see the right-angled triangle.
Use 22/7 only when the radius is a multiple of 7, and say so.
Write the part (arc, sector, segment) before computing.

Practice questions on Chord, Arc, Sector and Segment Problems

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 · MCQBoard level
The region between a chord and the arc it cuts off is called a:
(a) sector (b) segment (c) radius (d) diameter
Hint
Not the region between two radii.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
The diameter of a circle is 18 cm. State its radius and the length of the longest chord.
Hint
Longest chord is the diameter.
Answer
Radius $9$ cm; longest chord $18$ cm.
Q3 · 3 marks · Short answerHarder board level
A chord of length 16 cm is in a circle of radius 10 cm. Find its distance from the centre.
Hint
Half chord is 8.
Answer
$d = 1 0^{2} - 8^{2} = 36 = 6$ cm.
Q4 · 3 marks · Short answerHarder board level
A sector of a circle of radius 21 cm has a central angle of $6 0^{\circ}$ . Find its arc length and its perimeter.
Hint
Circumference first.
Answer
Circumference $= 2 \times \frac{22}{7} \times 21 = 132$ ; arc $= \frac{132}{6} = 22$ cm. Perimeter $= 22 + 21 + 21 = 64$ cm.
Q5 · 4 marks · Long answerHOTS / competency
Two parallel chords of lengths 6 cm and 8 cm are drawn in a circle of radius 5 cm. Find the distance between them if they are (i) on the same side of the centre, (ii) on opposite sides.
Hint
Find each distance from the centre.
Answer
Distances: for $6$ cm, $25 - 9 = 4$ ; for $8$ cm, $25 - 16 = 3$ . (i) Same side: $4 - 3 = 1$ cm. (ii) Opposite sides: $4 + 3 = 7$ cm.

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.Longest chord?Show answer
The diameter.
2.Circumference formula?Show answer
2 pi r.
3.Region between two radii and an arc?Show answer
A sector.
4.Semicircle perimeter?Show answer
pi r + 2r.

Chord, Arc, Sector and Segment Problems: frequently asked questions

Is a semicircle a sector?

Yes; it is a sector with a central angle of 180 degrees.

Is a segment the same as a sector?

No; a segment is cut off by a chord, a sector by two radii.