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CBSE · Class 10 · Mathematics

Areas Related to Circles 11.1: Areas of Sector and Segment of a Circle

5 concepts · 40 practice questions with answers · Free to read

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Sector, segment and the angle of a sector

From daily life

A slice of pizza is a sector: two straight cuts from the centre and the crust. A piece cut off by one straight knife line is a segment.

Simple definition

A sector is the region between two radii and the arc joining them; its angle at the centre is the central angle θ. A segment is the region between a chord and its arc. A sector with θ below 180° is minor, above 180° it is major, and together they fill 360°.

Oθsectorsegment

In the figure: Left: a sector with angle θ. Right: a segment cut off by a chord.

Rule 1: The angle of the other sector — Minor and major sector angles add up to 360°.

  • Easy: Minor sector 110°: major sector = 360° − 110° = 250°.
  • Medium: Major sector 300°: minor sector = 60°.
  • Exam-type: A semicircle is a sector of 180°.

Rule 2: Angle as a fraction of the circle — A sector of θ degrees is θ/360 of the whole circle.

  • Easy: 90° is 1/4 of the circle and 60° is 1/6.
  • Medium: 120° is 1/3; 72° is 72/360 = 1/5.
  • Exam-type: 45° is 1/8 of the circle.
Remember

Every sector and arc quantity is the same fraction θ/360 of the whole-circle quantity.

Common mistake

Using θ/180 instead of θ/360; the full turn is 360°.

Memory clue

"Angle over 360 is the slice of the whole."

Practice questions

Easy
  1. 1The angle of a minor sector of a circle is 110°. What is the angle of the corresponding major sector (in degrees)?
  2. 2A sector of a circle has an angle of 20° at the centre. What fraction of the area of the whole circle is the area of the sector?
  3. 3A sector of a circle has an angle of 10° at the centre. What fraction of the area of the whole circle is the area of the sector?
Medium
  1. 4The angle of a minor sector of a circle is 50°. What is the angle of the corresponding major sector (in degrees)?
  2. 5A sector of angle 40° is cut from a circle. Find the ratio of the area of this (minor) sector to the area of the corresponding major sector.
  3. 6A sector of angle 120° is cut from a circle. Find the ratio of the area of this (minor) sector to the area of the corresponding major sector.
Challenging
  1. 7A sector of angle 30° is cut from a circle. Find the ratio of the area of this (minor) sector to the area of the corresponding major sector.
  2. 8Two sectors have angles 72° and 40° and belong to circles of radii 5 cm and 8 cm. Find the ratio of their arc lengths.
Show answers (try them yourself first)
  1. 1250
  2. 21/18
  3. 31/36
  4. 4310
  5. 51 : 8
  6. 61 : 2
  7. 71 : 11
  8. 89 : 8

Length of an arc

From daily life

The tip of a clock’s minute hand travels along an arc. The more minutes pass, the longer the arc.

Simple definition

The arc of a sector with central angle θ is the fraction θ/360 of the circumference 2πr: l = θ/360 × 2πr. Here π is taken as 22/7 when r is a multiple of 7, otherwise 3.14.

Formula: l = θ/360 × 2πr

Oθrll = θ/360 × 2πrr = 21, θ = 60°l = 1/6 × 132= 22

In the figure: r = 21, θ = 60°: l = 1/6 × 132 = 22.

Rule 1: Find the arc length — First find the circumference, then take the fraction.

  • Easy: r = 21 cm, θ = 60°: circumference 132, l = 22 cm.
  • Medium: r = 7 cm, θ = 90°: circumference 44, l = 11 cm.
  • Exam-type: r = 9 cm, θ = 40°: l = 1/9 × 18π = 2π cm.

Rule 2: Find the angle — Turn the formula round: θ = l ÷ (2πr) × 360.

  • Easy: l = 22 cm, r = 21 cm: θ = 22/132 × 360 = 60°.
  • Medium: l = 11 cm, r = 7 cm: θ = 11/44 × 360 = 90°.
  • Exam-type: If l is the whole circumference, θ = 360°.
Remember

Arc length is a length, so its unit is cm or m, not cm².

Common mistake

Using πr² (the area formula) for the arc; the arc needs 2πr.

Memory clue

"Slice fraction times round-the-edge."

Practice questions

Easy
  1. 1Find the length of the arc (in cm, in terms of π) of a circle of radius 33 cm subtending an angle of 40° at the centre.
  2. 2Find the length of the arc (in cm, in terms of π) of a circle of radius 2 cm subtending an angle of 45° at the centre.
  3. 3Find the length of the arc (in cm, in terms of π) of a circle of radius 15 cm subtending an angle of 30° at the centre.
Medium
  1. 4Find the length of the arc (in cm, in terms of π) of a circle of radius 28 cm subtending an angle of 45° at the centre.
  2. 5Find the length of the arc (in cm, in terms of π) of a circle of radius 13 cm subtending an angle of 120° at the centre.
  3. 6Find the length of the arc (in cm, in terms of π) of a circle of radius 27 cm subtending an angle of 120° at the centre.
Challenging
  1. 7Find the length of the arc (in cm, in terms of π) of a circle of radius 18 cm subtending an angle of 72° at the centre.
  2. 8Find the length of the arc (in cm, in terms of π) of a circle of radius 11 cm subtending an angle of 40° at the centre.
Show answers (try them yourself first)
  1. 122π/3
  2. 2π/2
  3. 35π/2
  4. 47π
  5. 526π/3
  6. 618π
  7. 736π/5
  8. 822π/9

Area of a sector

From daily life

The patch of grass a goat can eat when tied at a corner of a field is a sector; its size depends on the rope length and the corner angle.

Simple definition

The area of a sector is the fraction θ/360 of the circle’s area πr²: A = θ/360 × πr². Using the arc length l, it is also A = ½ × l × r.

Formula: A = θ/360 × πr²; A = ½ l r

Oθ = 90°r = 14A = θ/360 × πr²= 1/4 × 22/7 × 196= 154

In the figure: r = 14 cm, θ = 90°: A = 1/4 × 22/7 × 196 = 154 cm².

Live figure

Area of a sector

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Rule 1: Find the area — Area of the whole circle first, then the fraction.

  • Easy: r = 14 cm, θ = 90°: A = 154 cm².
  • Medium: r = 21 cm, θ = 60°: A = 1/6 × 1386 = 231 cm².
  • Exam-type: r = 6 cm, θ = 60°: A = 1/6 × 36π = 6π cm².

Rule 2: Use ½ l r or find the angle — Both routes give the same area.

  • Easy: l = 22 cm, r = 21 cm: A = ½ × 22 × 21 = 231 cm².
  • Medium: A = 154 cm², r = 14 cm: θ = 154/616 × 360 = 90°.
  • Exam-type: A 10-minute sweep of a 21 cm minute hand is 60°: area 231 cm².
Remember

Check that the sector area is less than the whole circle’s area.

Common mistake

Using 2πr in the area formula instead of πr².

Memory clue

"Pie slice = fraction × pi r squared."

Practice questions

Easy
  1. 1Find the area (in cm², in terms of π) of the sector of angle 40° in a circle of radius 24 cm.
  2. 2Find the area (in cm², in terms of π) of the sector of angle 270° in a circle of radius 2 cm.
  3. 3Find the area (in cm², in terms of π) of the sector of angle 180° in a circle of radius 9 cm.
Medium
  1. 4Find the area (in cm², in terms of π) of the sector of angle 30° in a circle of radius 23 cm.
  2. 5Find the area (in cm², in terms of π) of the sector of angle 270° in a circle of radius 27 cm.
  3. 6The length of an arc of a circle of radius 28 cm is 39 cm. Find the area of the corresponding sector (in cm²).
Challenging
  1. 7The length of an arc of a circle of radius 5 cm is 37 cm. Find the area of the corresponding sector (in cm²).
  2. 8The length of an arc of a circle of radius 10 cm is 9 cm. Find the area of the corresponding sector (in cm²).
Show answers (try them yourself first)
  1. 164π
  2. 23π
  3. 381π/2
  4. 4529π/12
  5. 52187π/4
  6. 6546
  7. 792.5
  8. 845

Area of a segment

From daily life

To find the area of a lens-shaped window, cut away the straight-sided triangle from the sector that contains it.

Simple definition

A segment is a sector with a triangle removed: area of segment = area of sector − area of triangle. The triangle has two sides r and angle θ between them. For 90° its area is ½r²; for 60° it is an equilateral triangle with area (√3/4) r²; for 120° it is (√3/4) r² again.

Formula: segment = θ/360 × πr² − triangle

sectortrianglesegment−=segment = sector − triangler = 10, θ = 90°: 78.5 − 50 = 28.5

In the figure: r = 10, θ = 90°: sector 78.5 − triangle 50 = segment 28.5 cm² (π = 3.14).

Live figure

Area of a segment

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Rule 1: Right-angle and 60° segments — Match the triangle to the angle.

  • Easy: r = 10, θ = 90°: 78.5 − 50 = 28.5 cm².
  • Medium: r = 12, θ = 60°: sector 75.36; triangle (√3/4) × 144 = 62.28; segment 13.08 cm² (π = 3.14, √3 = 1.73).
  • Exam-type: A segment is always smaller than its sector.

Rule 2: A 120° segment — Use half the product of two sides times sin θ.

  • Easy: r = 6, θ = 120°: sector = 1/3 × 3.14 × 36 = 37.68 cm².
  • Medium: Triangle = ½ × 6 × 6 × sin 120° = 9√3 ≈ 15.57 cm².
  • Exam-type: Segment = 37.68 − 15.57 = 22.11 cm².
Remember

Draw the sector, the triangle and the segment as three separate pictures before subtracting.

Common mistake

Adding the triangle to the sector; the segment is the sector minus the triangle for a minor segment.

Memory clue

"Slice minus triangle leaves the segment."

Practice questions

Easy
  1. 1In a circle with centre O, the area of the sector OAPB is 372 cm² and the area of the triangle OAB is 96 cm². Find the area of the minor segment APB (in cm²).
  2. 2A circle has area 193 cm² and its minor segment has area 24 cm². Find the area of the corresponding major segment (in cm²).
  3. 3In a circle with centre O, the area of the sector OAPB is 102 cm² and the area of the triangle OAB is 53 cm². Find the area of the minor segment APB (in cm²).
Medium
  1. 4In a circle with centre O, the area of the sector OAPB is 388 cm² and the area of the triangle OAB is 369 cm². Find the area of the minor segment APB (in cm²).
  2. 5A chord of a circle of radius 18 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment (in cm², in terms of π and √3).
  3. 6A chord of a circle of radius 12 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment (in cm², in terms of π and √3).
Challenging
  1. 7A chord of a circle of radius 48 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment (in cm², in terms of π and √3).
  2. 8A chord of a circle of radius 4 cm subtends 90° at the centre. Find the area of the corresponding major segment (in cm², in terms of π).
Show answers (try them yourself first)
  1. 1276
  2. 2169
  3. 349
  4. 419
  5. 554π − 81√3
  6. 624π − 36√3
  7. 7384π − 576√3
  8. 812π + 8

Combined figures and applications

From daily life

A tethered horse, a wiper blade and a clock hand are all sectors in real life; garden beds with rounded corners combine squares and quarter circles.

Simple definition

For a combined figure, split it into squares, rectangles, triangles, sectors and segments, find each area, and then add or subtract. A rope fixed at a corner sweeps a sector whose angle is the corner’s angle.

1414r = 7shaded =square − circle= 196 − 154= 42

In the figure: Square of side 14 with four quarter circles of radius 7 cut out: shaded area = 196 − 154 = 42.

Rule 1: Rope and wiper problems — Find the angle swept, then use the sector area.

  • Easy: A horse tied at a field corner with a 14 m rope covers 1/4 × 22/7 × 196 = 154 m².
  • Medium: A 21 cm wiper blade sweeping 120° covers 1/3 × 1386 = 462 cm².
  • Exam-type: The 21 cm minute hand in 10 minutes: area 231 cm², tip travels 22 cm.

Rule 2: Square with quarter circles — Four quarter circles of equal radius add up to one full circle.

  • Easy: Square 14 cm, quarter circles r = 7: shaded = 196 − 154 = 42 cm².
  • Medium: Check: four quarters of 154/4 = 38.5 give 154.
  • Exam-type: Perimeter of the inner shape is the full circle 44 cm.
Remember

Always ask: which smaller shapes build this figure, and do I add or subtract them?

Common mistake

Using the full circle’s area when only a corner sector is reachable by the rope.

Memory clue

"Split, find each, then add or subtract."

Practice questions

Easy
  1. 1Find the area of a quadrant of a circle of radius 49 cm (in cm²). (Use π = 22/7.)
  2. 2Find the area of a quadrant of a circle of radius 42 cm (in cm²). (Use π = 22/7.)
  3. 3Find the perimeter of a sector of a circle of radius 20 cm and angle 270° (in cm). (Use π = 3.14.)
Medium
  1. 4Find the area of a semicircle of radius 28 cm (in cm²). (Use π = 22/7.)
  2. 5Find the perimeter of a sector of a circle of radius 50 cm and angle 36° (in cm). (Use π = 3.14.)
  3. 6Find the area of a semicircle of radius 7 cm (in cm²). (Use π = 22/7.)
Challenging
  1. 7Find the perimeter of a sector of a circle of radius 7 cm and angle 90° (in cm). (Use π = 22/7.)
  2. 8Find the perimeter of a sector of a circle of radius 21 cm and angle 90° (in cm). (Use π = 22/7.)
Show answers (try them yourself first)
  1. 11886.5
  2. 21386
  3. 3134.2
  4. 41232
  5. 5131.4
  6. 677
  7. 725
  8. 875