Area of Sector of a Circle
The Area of a Sector is a key topic in CBSE Class 10 Mathematics, Chapter 12 (Areas Related to Circles). A sector is the region enclosed between two radii and the arc of a circle — like a "slice of pizza."
There are two types of sectors:
- Minor sector — the smaller region (angle < 180°).
- Major sector — the larger region (angle > 180°).
The area of a sector depends on two quantities: the radius of the circle and the central angle (the angle subtended by the arc at the centre). The formula is a proportional part of the total area of the circle.
What is Area of Sector of a Circle - Formula, Derivation & Solved Examples?
Definition: A sector of a circle is the region bounded by two radii and the arc intercepted between them.
Types of sectors:
- Minor sector: The sector with the smaller area. Its central angle θ < 180°.
- Major sector: The sector with the larger area. Its central angle = 360° − θ.
Related terms:
- Central angle (θ) — the angle formed at the centre by the two radii.
- Arc — the curved part of the circle that forms one boundary of the sector.
- Chord — the straight line joining the endpoints of the arc.
- Segment — the region between a chord and the arc. (Sector = Segment + Triangle)
Relationship:
- Area of minor sector + Area of major sector = Area of the full circle (πr²).
- Area of segment = Area of sector − Area of the triangle formed by the two radii and the chord.
Area of Sector of a Circle Formula
Area of Sector:
Area of Sector = (θ/360°) × πr²
Where:
- θ = central angle of the sector (in degrees)
- r = radius of the circle
- π = 22/7 or 3.14159...
Area of Major Sector:
Area of Major Sector = ((360° − θ)/360°) × πr²
Or equivalently:
Area of Major Sector = πr² − Area of Minor Sector
Arc Length of a Sector:
Arc Length = (θ/360°) × 2πr
Perimeter of a Sector:
Perimeter = 2r + Arc Length = 2r + (θ/360°) × 2πr
Derivation and Proof
Derivation of Sector Area Formula:
- The area of a full circle with radius r is πr².
- A full circle subtends an angle of 360° at the centre.
- If the entire 360° gives area πr², then 1° gives area πr²/360.
- Therefore, an angle of θ degrees gives area = (θ/360°) × πr².
Derivation of Arc Length Formula:
- The circumference (full arc) of a circle is 2πr.
- The full circle corresponds to 360°.
- An arc subtending angle θ at the centre has length = (θ/360°) × 2πr.
Alternative form using arc length:
- Let arc length = l. Then l = (θ/360°) × 2πr.
- Area of sector = (1/2) × l × r = (1/2) × r × (θ/360°) × 2πr = (θ/360°) × πr².
- This is analogous to the area of a triangle = (1/2) × base × height, where the "base" is the arc and the "height" is the radius.
Types and Properties
Problems on area of sector in Class 10 include:
Type 1: Direct Calculation of Sector Area
- Given θ and r, find the area using (θ/360°) × πr².
Type 2: Finding Arc Length
- Given θ and r, find arc length using (θ/360°) × 2πr.
Type 3: Finding Major Sector Area
- Area of major sector = πr² − area of minor sector.
Type 4: Finding Area of Segment
- Area of segment = Area of sector − Area of triangle.
- Requires knowledge of area of triangle using (1/2)r²sin θ or standard triangle formulas.
Type 5: Combination Figures
- Shaded regions formed by overlapping sectors, sectors cut from rectangles, etc.
Type 6: Finding Angle Given Area or Arc Length
- Reverse problems: given sector area and radius, find θ.
Type 7: Perimeter of a Sector
- Perimeter = 2r + arc length.
Methods
Method 1: Direct Formula Application
- Identify the central angle θ (in degrees).
- Identify the radius r.
- Substitute into: Area = (θ/360°) × πr².
- Use π = 22/7 or 3.14 as specified.
Method 2: Finding Segment Area
- Calculate sector area = (θ/360°) × πr².
- Calculate the area of the triangle formed by two radii and the chord.
- For common angles:
- θ = 60°: Equilateral triangle with side r → Area = (√3/4)r².
- θ = 90°: Right isosceles triangle → Area = (1/2)r².
- θ = 120°: Area = (√3/4)r².
- Segment area = Sector area − Triangle area.
Method 3: Reverse Calculation (Finding θ)
- Given: Area of sector = A, radius = r.
- A = (θ/360°) × πr².
- θ = (A × 360°)/(πr²).
Standard Triangle Areas (for segment problems):
| Central Angle θ | Triangle Area |
|---|---|
| 60° | (√3/4)r² |
| 90° | (1/2)r² |
| 120° | (√3/4)r² |
Solved Examples
Example 1: Finding Area of a Sector
Problem: Find the area of a sector with central angle 60° and radius 14 cm. (Use π = 22/7)
Solution:
Given:
- θ = 60°
- r = 14 cm
Using the formula:
- Area = (θ/360°) × πr²
- = (60/360) × (22/7) × 14²
- = (1/6) × (22/7) × 196
- = (1/6) × 22 × 28
- = (1/6) × 616
- = 102.67 cm²
Answer: The area of the sector is 102.67 cm².
Example 2: Finding Arc Length
Problem: Find the arc length of a sector with central angle 90° and radius 21 cm. (Use π = 22/7)
Solution:
Given:
- θ = 90°
- r = 21 cm
Using the formula:
- Arc length = (θ/360°) × 2πr
- = (90/360) × 2 × (22/7) × 21
- = (1/4) × 2 × 22 × 3
- = (1/4) × 132
- = 33 cm
Answer: The arc length is 33 cm.
Example 3: Finding Area of Minor and Major Sectors
Problem: A sector has central angle 120° in a circle of radius 21 cm. Find the areas of both the minor and major sectors. (Use π = 22/7)
Solution:
Given:
- θ = 120°, r = 21 cm
Area of minor sector:
- = (120/360) × (22/7) × 21²
- = (1/3) × (22/7) × 441
- = (1/3) × 22 × 63
- = (1/3) × 1386 = 462 cm²
Area of major sector:
- = πr² − 462
- = (22/7) × 441 − 462
- = 1386 − 462 = 924 cm²
Answer: Minor sector = 462 cm², Major sector = 924 cm².
Example 4: Finding Area of a Segment (90° sector)
Problem: Find the area of the minor segment of a circle with radius 14 cm and sector angle 90°. (Use π = 22/7)
Solution:
Given:
- θ = 90°, r = 14 cm
Area of sector:
- = (90/360) × (22/7) × 14² = (1/4) × (22/7) × 196 = (1/4) × 616 = 154 cm²
Area of triangle (right isosceles, two sides = r):
- = (1/2) × r × r = (1/2) × 14 × 14 = 98 cm²
Area of segment:
- = 154 − 98 = 56 cm²
Answer: The area of the minor segment is 56 cm².
Example 5: Finding Area of a Segment (60° sector)
Problem: A chord subtends an angle of 60° at the centre of a circle of radius 42 cm. Find the area of the minor segment. (Use π = 22/7)
Solution:
Given:
- θ = 60°, r = 42 cm
Area of sector:
- = (60/360) × (22/7) × 42² = (1/6) × (22/7) × 1764 = (1/6) × 5544 = 924 cm²
Area of triangle (equilateral, since two sides = r and included angle = 60°):
- = (√3/4) × r² = (√3/4) × 1764 = 441√3 ≈ 441 × 1.732 = 763.81 cm²
Area of segment:
- = 924 − 763.81 = 160.19 cm²
Answer: The area of the minor segment is (924 − 441√3) cm² ≈ 160.19 cm².
Example 6: Finding Central Angle from Sector Area
Problem: The area of a sector is 154 cm² and the radius is 14 cm. Find the central angle. (Use π = 22/7)
Solution:
Given:
- Area = 154 cm², r = 14 cm
Using Area = (θ/360°) × πr²:
- 154 = (θ/360) × (22/7) × 196
- 154 = (θ/360) × 616
- θ/360 = 154/616 = 1/4
- θ = 360/4 = 90°
Answer: The central angle is 90°.
Example 7: Perimeter of a Sector
Problem: Find the perimeter of a sector of a circle with radius 7 cm and central angle 60°. (Use π = 22/7)
Solution:
Given:
- r = 7 cm, θ = 60°
Arc length:
- = (60/360) × 2 × (22/7) × 7 = (1/6) × 44 = 22/3 cm
Perimeter of sector:
- = 2r + arc length = 14 + 22/3 = 42/3 + 22/3 = 64/3
- ≈ 21.33 cm
Answer: The perimeter is 64/3 cm ≈ 21.33 cm.
Example 8: Shaded Region Between Two Sectors
Problem: Two concentric circles have radii 7 cm and 14 cm. Find the area of the shaded region between the two sectors with central angle 45°. (Use π = 22/7)
Solution:
Given:
- r₁ = 7 cm (inner), r₂ = 14 cm (outer), θ = 45°
Area of outer sector:
- = (45/360) × (22/7) × 14² = (1/8) × 616 = 77 cm²
Area of inner sector:
- = (45/360) × (22/7) × 7² = (1/8) × 154 = 19.25 cm²
Shaded area:
- = 77 − 19.25 = 57.75 cm²
Answer: The shaded area is 57.75 cm².
Real-World Applications
Food Industry:
- Calculating the area of a pizza slice or a pie slice — each slice is a sector of a circle.
Architecture:
- Fan windows, sector-shaped floor tiles, and arc-shaped garden plots require sector area calculations.
Agriculture:
- Circular irrigation systems (centre-pivot irrigation) create sector-shaped irrigated areas when not completing a full circle.
Engineering:
- Gear teeth, cam profiles, and turbine blade cross-sections involve sector geometry.
Navigation:
- Radar systems scan sector-shaped regions. The area covered in one sweep is a sector of the radar's range circle.
Clock Problems:
- The area swept by a clock's minute or hour hand in a given time is a sector area calculation.
Key Points to Remember
- A sector is the region between two radii and an arc of a circle.
- Area of sector = (θ/360°) × πr², where θ is the central angle in degrees.
- Arc length = (θ/360°) × 2πr.
- Perimeter of sector = 2r + arc length.
- Minor sector has θ < 180°; major sector has angle = 360° − θ.
- Area of minor sector + Area of major sector = πr² (total circle area).
- Area of segment = Area of sector − Area of the triangle formed by two radii and the chord.
- For θ = 60°, the triangle is equilateral with area (√3/4)r².
- For θ = 90°, the triangle is right isosceles with area (1/2)r².
- The formula is based on proportionality: the sector's angle is the fraction θ/360° of the full circle.
Practice Problems
- Find the area of a sector with radius 28 cm and central angle 45°. (Use π = 22/7)
- The minute hand of a clock is 7 cm long. Find the area swept by it in 20 minutes.
- Find the area of the minor segment of a circle with radius 21 cm and sector angle 120°.
- A sector has arc length 22 cm and radius 21 cm. Find the central angle.
- Find the area of the shaded region between two concentric circles of radii 10.5 cm and 3.5 cm, with central angle 60°.
- The perimeter of a sector is 31.4 cm. If the radius is 7 cm, find the central angle. (Use π = 3.14)
Frequently Asked Questions
Q1. What is a sector of a circle?
A sector is the region enclosed between two radii and the intercepted arc of a circle. It is shaped like a slice of pizza.
Q2. What is the formula for area of a sector?
Area of sector = (θ/360°) × πr², where θ is the central angle in degrees and r is the radius.
Q3. What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc. A segment is bounded by a chord and an arc. Area of segment = Area of sector − Area of the triangle formed by the chord and the two radii.
Q4. How do you find the area of a major sector?
Area of major sector = πr² − Area of minor sector. Alternatively, use (360° − θ)/360° × πr², where θ is the minor sector's angle.
Q5. What is the arc length formula?
Arc length = (θ/360°) × 2πr. It is the proportional part of the circumference corresponding to the central angle θ.
Q6. How is the sector area formula derived?
A full circle has area πr² for 360°. By proportion, an angle θ gives area = (θ/360°) × πr². The sector area is directly proportional to the central angle.
Q7. What is the perimeter of a sector?
Perimeter of a sector = 2r + arc length = 2r + (θ/360°) × 2πr. It includes the two straight sides (radii) and the curved part (arc).
Q8. How is the area of a segment related to the area of a sector?
Area of segment = Area of sector − Area of the triangle formed by the two radii and the chord. For a 90° sector, triangle area = (1/2)r². For a 60° sector, triangle area = (√3/4)r².
