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State Board · Class 8 · Mathematics

Lesson 17: Circle: Chord and Arc

4 concepts · 32 practice questions with answers · Free to read

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Chord and parts of a circle

From daily life

The hub of a cycle wheel is the centre. A wire joining two points on the rim is a chord. If it passes through the centre it is the longest chord — the diameter.

Simple definition

A segment joining any two points on a circle is a chord. The diameter = 2 × radius is the longest chord.

1058

In the figure: Radius 5 (green), diameter 10 (orange) is the longest chord, and a chord of 8 (blue): 8 < 10.

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Chord and parts of a circle

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  • Easy: r = 7 cm ⇒ diameter = 2 × 7 = 14 cm.
  • Medium: Diameter 18 cm ⇒ r = 9; a chord of 15 cm is possible (15 ≤ 18), a chord of 20 cm is not.
  • Exam-type: In a circle with r = 6, the chord 12 is the diameter; chords 7 and 10 are shorter; the longest chord is 12.
Remember

Every diameter is a chord, but not every chord is a diameter.

Common mistake

Treating chord and diameter as the same. The diameter is only the chord through the centre.

Shortcut

A chord longer than the diameter is impossible. Memory clue: The diameter is the longest chord: "chord ≤ diameter".

Spot the difference

PointChordDiameter
Passes through centreusually notyes
Lengthshorter than the diameterlongest, 2 × r
How manymanymany, all equal to 2r
Example (r = 5)810

Practice questions

Easy
  1. 1If r = 9 cm, what is the diameter?
  2. 2If the diameter is 26 cm, what is the radius?
  3. 3Which is the longest chord of a circle?
Medium
  1. 4Is a chord of 15 cm possible in a circle with r = 8?
  2. 5Is a chord of 17 cm possible in a circle with r = 8?
  3. 6The diameter of a circle is 30 cm. Chords are 12, 25, 30: which one is the diameter?
Challenging
  1. 7A chord is 14 cm and the radius is 7 cm. Is the chord a diameter?
  2. 8A circle with r = 10 has chords of 10, 16 and 20 cm. Which chords are possible?
Show answers (try them yourself first)
  1. 12 × 9 = 18 cm.
  2. 226 ÷ 2 = 13 cm.
  3. 3The diameter.
  4. 4Diameter 16; 15 ≤ 16, so it is possible.
  5. 5Diameter 16; 17 > 16, so it is not possible.
  6. 6The 30 cm chord, because it is the longest.
  7. 7Diameter = 2 × 7 = 14; yes, the chord is a diameter.
  8. 8Diameter 20; all three (10, 16, 20) are ≤ 20, so they are possible; 20 is the diameter.

Perpendicular from the centre to a chord

From daily life

A plank lies across a round pit. A rope dropped from the middle of the plank passes through the centre and meets the plank at a right angle.

Simple definition

The perpendicular from the centre to a chord bisects the chord. Conversely, the line from the centre to the midpoint of a chord is perpendicular to it.

61088

In the figure: Chord 16, radius 10, distance from the centre 6: half-chord 8; 6² + 8² = 10² (right triangle).

Formula: (radius)² = (distance)² + (half chord)²

In words: Centre O, chord PQ, midpoint M: OM, MP and OP form a right triangle (hypotenuse = radius).

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Perpendicular from the centre to a chord

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  • Easy: r = 10, distance from centre 6: half chord = √(10² − 6²) = √64 = 8 ; chord = 16.
  • Medium: Chord 24, r = 13: half chord 12 ; distance = √(13² − 12²) = √25 = 5.
  • Exam-type: Chord 48, r = 25: half chord 24 ; distance = √(25² − 24²) = √49 = 7.
Remember

Half-chord, distance and radius always form a right triangle; the hypotenuse is the radius.

Common mistake

Always measure the distance along the perpendicular, not along a slanted line.

Shortcut

The nearer a chord is to the centre, the longer it is. The diameter is at distance 0. Memory clue: A perpendicular is a bisector: the perpendicular from the centre cuts the chord in half.

Spot the difference

PointChord lengthDistance from centre
If the chord is near the centrelongersmaller
Longest chorddiameter0 (through the centre)
In r = 10166
How to find2 × √(r² − d²)√(r² − (half-chord)²)

Practice questions

Easy
  1. 1r = 5, distance from the centre 3. Length of the chord?
  2. 2r = 13, distance 5. Chord?
  3. 3Chord 16, r = 10. Distance from the centre?
Medium
  1. 4Chord 30, r = 17. Distance?
  2. 5Distance from the centre 12 and chord 10. Radius?
  3. 6Chord 18 and distance from the centre 12. Radius?
Challenging
  1. 7In a circle with r = 10 there are chords of 12 cm and 16 cm. Which is nearer the centre and why?
  2. 8Two equal chords of 24 each lie in a circle with r = 15. Find their distances from the centre and note what you see.
Show answers (try them yourself first)
  1. 1Half-chord = √(25 − 9) = 4; chord = 8.
  2. 2Half-chord = √(169 − 25) = 12; chord = 24.
  3. 3Half-chord 8; distance = √(100 − 64) = 6.
  4. 4Half-chord 15; distance = √(289 − 225) = 8.
  5. 5Half-chord 5; r = √(144 + 25) = 13.
  6. 6Half-chord 9; r = √(144 + 81) = √225 = 15.
  7. 7Distance of 12 = √(100 − 36) = 8; distance of 16 = √(100 − 64) = 6; the 16 chord is nearer, because a longer chord is nearer the centre.
  8. 8Each distance = √(225 − 144) = 9; equal chords are at equal distances from the centre.

Arc and central angle

From daily life

A pizza slice has its tip at the centre and its crust is the arc. The wider the slice, the bigger the angle.

Simple definition

The part of a circle between two points is an arc. The small part is the minor arc, the big part the major arc. The central angle equals the measure of the minor arc.

70°290°

In the figure: Central angle 70°: minor arc 70° (orange), major arc 360° − 70° = 290° (blue).

Formula: Major arc = 360° − minor arc ; semicircle = 180°

Live figure

Arc and central angle

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  • Easy: Central angle 70°: minor arc 70° ; major arc 360 − 70 = 290°.
  • Medium: Major arc is twice the minor arc. x + 2x = 360 ; x = 120 ; minor arc 120°, major arc 240°.
  • Exam-type: Major arc exceeds the minor arc by 100°: 2x + 100 = 360 ; x = 130 ; minor arc 130°, major arc 230°.
Remember

The minor arc is always less than 180°, the major arc more than 180° (a semicircle is exactly 180°).

Common mistake

Taking the major arc equal to the central angle. The central angle equals the minor arc.

Shortcut

Minor arc = central angle; major arc = 360° − central angle. Memory clue: Minor arc = the angle, major arc = the rest; together they make the full circle.

Spot the difference

PointMinor arcMajor arc
Measureless than 180°more than 180°
Formula= central angle360° − central angle
If the angle is 70°70°290°
Semicircle180°180°

Practice questions

Easy
  1. 1Central angle 50°. Minor arc and major arc?
  2. 2Minor arc 110°. Major arc?
  3. 3What is the measure of a semicircle?
Medium
  1. 4Major arc 300°. Central angle?
  2. 5The major arc is three times the minor arc. Find both arcs.
  3. 6The major arc is 60° more than the minor arc. Find both arcs.
Challenging
  1. 7In a clock, find the central angle and minor arc between 12 and 4.
  2. 8One arc measures 135°. How much is the other arc, and which is minor and which is major?
Show answers (try them yourself first)
  1. 1Minor arc 50°; major arc 360 − 50 = 310°.
  2. 2360 − 110 = 250°.
  3. 3180°.
  4. 4Minor arc = 360 − 300 = 60°; central angle 60°.
  5. 5x + 3x = 360; x = 90; minor arc 90°, major arc 270°.
  6. 62x + 60 = 360; x = 150; minor arc 150°, major arc 210°.
  7. 7Each hour is 30°; 4 × 30 = 120°; minor arc 120°.
  8. 8The other arc = 360 − 135 = 225°; 135° is the minor arc and 225° the major arc.

Equal arcs and equal chords

From daily life

On the same clock the stretch from 12 to 2 and from 4 to 6 are equal arcs, so the chords joining their ends are equal too.

Simple definition

In the same circle, equal arcs have equal chords and equal chords cut equal arcs. Equal chords are at equal distance from the centre.

65°65°

In the figure: Both central angles are 65° ⇒ equal arcs ⇒ equal chords (same tick marks).

Formula: ∠AOB = ∠COD ⇒ arc AB = arc CD ⇒ chord AB = chord CD

Live figure

Equal arcs and equal chords

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  • Easy: Chord AB = chord CD and ∠AOB = 65°. So ∠COD = 65° and arc AB = arc CD = 65°.
  • Medium: Two equal chords of 16 and 16 (r = 10). Both are at distance √(100 − 64) = 6 from the centre.
  • Exam-type: ∠AOB = ∠COD = 80°; arc AB = 80°, arc CD = 80°; if chord AB = 12 then chord CD = 12.
Remember

In the same circle: equal arcs ⇔ equal chords ⇔ equal central angles.

Common mistake

This holds only in the same circle or in equal circles, not in circles of different radii.

Shortcut

Once one of the three is equal, the other two are equal automatically. Memory clue: Equal arcs ⇔ equal chords ⇔ equal angles: "give one, get three".

Spot the difference

PointChordArc
Shapestraight segmentcurved part
Measurelength in cmin degrees (equal to central angle)
Equalityequal arcs ⇒ equal chordsequal chords ⇒ equal arcs

Practice questions

Easy
  1. 1r = 5 cm. What is the longest chord?
  2. 2r = 10, distance from centre 6. Chord?
  3. 3Minor arc 80°. Major arc?
Medium
  1. 4Chord 24, r = 13. Distance from the centre?
  2. 5Chord 10, r = 13. Distance from the centre?
  3. 6Chord AB = chord CD and ∠AOB = 50°. Find ∠COD and arc CD.
Challenging
  1. 7Two parallel chords of length 6 and 8 in a circle with r = 5. Find the gap between them if on the same side of the centre; and if on opposite sides.
  2. 8The major arc is 100° more than the minor arc. Find both arcs.
Show answers (try them yourself first)
  1. 1Diameter = 2 × 5 = 10 cm.
  2. 22 × √(100 − 36) = 2 × 8 = 16.
  3. 3360 − 80 = 280°.
  4. 4√(169 − 144) = 5.
  5. 5Half chord 5 ; distance = √(169 − 25) = √144 = 12.
  6. 6∠COD = 50° ; arc CD = 50°.
  7. 7Chord 6: distance √(25 − 9) = 4 ; chord 8: distance √(25 − 16) = 3. Same side: 4 − 3 = 1 ; opposite sides: 4 + 3 = 7.
  8. 82x + 100 = 360 ; x = 130 ; minor arc 130°, major arc 230°.