Areas Related to Circles — Class 10 Mathematics
"Pi (π) — the mathematics of circles, an irrational number with infinite digits, present everywhere."
1. About the Chapter
This chapter expands circle-area calculations to include:
- Sector (pie-slice region)
- Segment (region between chord and arc)
- Combinations of plane figures (composite shapes)
Foundation for engineering, design, daily calculations.
2. Basic Circle Formulas (Recap)
Circumference
C = 2πr (where r = radius) C = πd (where d = diameter = 2r)
Area
A = πr²
Value of π
- π ≈ 3.14159...
- For class use: π = 22/7 (when problems give r as multiple of 7)
- Or: π = 3.14 (decimal approximation)
3. Sector of a Circle
Definition
A sector is a region between two radii and an arc — like a pie slice.
Angle of Sector
The angle between the two radii (let it be θ).
Sector Area
Area of sector = (θ/360°) × πr²
(Since full circle is 360°, sector at angle θ is fraction θ/360°.)
Length of Arc
Arc length = (θ/360°) × 2πr
(Fraction of total circumference.)
Example
A sector with radius 14 cm and angle 90° (quarter circle).
- Area = (90/360) × 22/7 × 14² = (1/4) × 22/7 × 196 = 154 cm²
- Arc length = (90/360) × 2 × 22/7 × 14 = (1/4) × 88 = 22 cm
4. Segment of a Circle
Definition
A segment is a region between a CHORD and the corresponding ARC.
Types
- Minor segment: smaller region (less than half circle)
- Major segment: larger region (more than half circle)
Segment Area
Area of MINOR segment = Area of sector − Area of triangle
(The triangle is formed by the two radii and the chord.)
For sector angle θ: Area of segment = (θ/360°) × πr² − (1/2) r² sin θ
(The second term is area of triangle using ½ × base × height, or for any triangle, ½ ab sin C.)
Example
Find area of segment of circle (radius 10, angle 90°).
- Sector area = (90/360) × π × 100 = 25π ≈ 78.5 cm²
- Triangle area = (1/2)(10)(10) sin 90° = 50 cm²
- Segment area = 78.5 − 50 = 28.5 cm²
Major Segment
- Major segment = Area of circle − Minor segment
- = πr² − Minor segment
5. Combinations of Plane Figures
Strategy
For complex shapes:
- DIVIDE into known simple shapes (rectangle, circle, triangle, etc.)
- CALCULATE each part's area
- ADD (for combined area) or SUBTRACT (for cut-out shapes)
Example 1: Park
A rectangular park 50 m × 30 m has a semicircular flower bed of radius 10 m at one end.
- Park area = 50 × 30 = 1500 m²
- Flower bed area = (1/2)π(10)² = 50π ≈ 157 m²
- Lawn area = 1500 − 157 = 1343 m²
Example 2: Window
A rectangular window 1.5 m wide with a semicircular top of radius 0.75 m. Find area for glass needed.
- Rectangular part: assume height = 2 m. Area = 1.5 × 2 = 3 m²
- Semicircle: (1/2)π(0.75)² ≈ 0.88 m²
- Total area ≈ 3.88 m²
Example 3: Hollow Pipe
A pipe has outer radius 5 cm, inner radius 3 cm. Cross-section area?
- Outer circle area = π(5)² = 25π
- Inner circle area = π(3)² = 9π
- Cross-section = 25π − 9π = 16π ≈ 50.27 cm²
6. Worked Examples
Example 1: Find Arc Length
A circle has radius 21 cm. Find arc length subtending 60° at centre.
- Arc length = (60/360) × 2 × 22/7 × 21
- = (1/6) × 132 = 22 cm
Example 2: Find Sector Angle
Sector of radius 10 has arc length of 5.6π. Find angle.
- 5.6π = (θ/360) × 2π × 10
- θ = (5.6 × 360) / 20 = 100.8°
Example 3: Cut from Square
A square of side 14 cm has 4 semicircles drawn on its sides. Find the area of the shape formed.
- Square area = 14² = 196 cm²
- 4 semicircles = 2 full circles of radius 7
- Total = 2 × π × 49 = 98π ≈ 308 cm²
- Net area = 196 + 308 = 504 cm² (if added) OR 196 − 308 = NEGATIVE (so different setup)
(Actual setup depends — if circles are added outside or are cut from corners.)
Example 4: Two Wheels
A wheel has diameter 56 cm. How many revolutions for 11 km?
- Circumference = π × 56 = (22/7) × 56 = 176 cm = 1.76 m
- Distance = 11 km = 11,000 m
- Revolutions = 11,000 / 1.76 = 6,250
7. Common Mistakes
-
Confusing sector with segment
- Sector = between TWO RADII and arc.
- Segment = between CHORD and arc.
-
Wrong angle conversion
- Always use angle out of 360° in fraction.
-
Wrong π value
- Use 22/7 if numbers are multiples of 7; else 3.14.
-
Forgetting to convert units
- cm² vs m². 1 m² = 10,000 cm².
-
Missing parts in combinations
- Check if shape adds or subtracts. Re-read problem.
8. Real-World Applications
Architecture
- Domed buildings: surface area
- Stadiums: layout calculations
- Indian temples use circle geometry
Engineering
- Wheel/cog calculations
- Pipe cross-sections
- Pizza box, soup can design
Design
- Logos, watch faces
- Sport fields (cricket field with rope, football pitch curves)
Indian Use
- Indian Rangoli patterns use sectors and segments
- Sundials use sector geometry
- Cricket boundary calculations
9. Indian Context
π Approximations Through Indian History
- Aryabhata (5th c.): π ≈ 3.1416 (very accurate)
- Bhaskara II: refined π calculations
- Madhava (14th c.): infinite series for π — 200 years before Newton!
This made India a global leader in circle geometry.
10. Conclusion
Areas related to circles bring together:
- Pi (π)
- Geometry
- Algebra
Master:
- Sector area: (θ/360°) × πr²
- Arc length: (θ/360°) × 2πr
- Segment = sector − triangle
- Combinations: ADD/SUBTRACT simple shapes
Practice 15+ problems. This chapter combines Chapter 10 (circles), Chapter 12 (volumes), and basic geometry.
Circles and their parts — the geometry that surrounds us.
