Areas Related to Circles — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Areas Related to Circles (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Areas Related to Circles (CBSE & HBSE)
Areas Related to Circles combines geometry and mensuration. CBSE tests sector, segment, and combination areas. HBSE focuses on direct formula application with standard figures.
Sector and Segment of a Circle
Parts of a Circle
| Term | Definition | Formula |
|---|---|---|
| Sector | Region between two radii and an arc | (θ/360°) × πr² |
| Minor Sector | Sector with smaller arc | Same formula, θ < 180° |
| Major Sector | Sector with larger arc | (360°-θ)/360° × πr² |
| Segment | Region between chord and arc | Area(sector) - Area(triangle) |
| Minor Segment | Region of minor arc side | Sector area - triangle area |
| Major Segment | Region of major arc side | Circle area - minor segment |
Key Formulas
$$\text{Area of Sector} = \frac{\theta}{360°} \times \pi r^2$$
$$\text{Length of Arc} = \frac{\theta}{360°} \times 2\pi r$$
$$\text{Perimeter of Sector} = 2r + \text{arc length} = 2r + \frac{\theta}{360°} \times 2\pi r$$
$$\text{Area of Minor Segment} = \text{Area of Sector} - \text{Area of Triangle}$$
Area of Triangle for Common Angles
| θ | Triangle formula |
|---|---|
| 90° | (1/2)r² |
| 60° | (√3/4)r² |
| 120° | (√3/4)r² (same as 60° for equilateral properties) |
IMPORTANT: Use π = 22/7 when radius is a multiple of 7. Use π = 3.14 otherwise (as directed by question).
Relationship: Sector → Cone
When a sector of radius l and angle θ is rolled into a cone:
- Slant height of cone = l
- Base circumference of cone = arc length = (θ/360°) × 2πl
- Radius of cone base = (θ/360°) × l
Areas of Combinations of Figures
Common Combination Problems
Type 1: Square/Rectangle with inscribed/excircle
- Circle inscribed in square of side a: radius = a/2
- Circle circumscribed about square of side a: radius = a/√2 = a√2/2
- Shaded area = Area(square) - Area(circle) OR Area(circle) - Area(square)
Type 2: Flower/Petal designs
- Two semicircles or sectors overlapping
- Area of 'petal' (vesica piscis) = 2 × sector area - rhombus area
Type 3: Race track / Ring $$\text{Area of ring} = \pi(R^2 - r^2) = \pi(R+r)(R-r)$$ where R = outer radius, r = inner radius
Type 4: Shaded region in circle
- Find area of whole circle, subtract unshaded regions
- Or: add individual shaded regions
Standard Questions
Segment shaded: Area = (θ/360°)πr² - (1/2)r²sin θ
Four quarter circles in corners of rectangle: Area of 4 quarter circles = π × r² (if all same radius r)
Horse grazing problems: "A horse tied at corner of square field of side a, with rope of length r. Find grazing area."
- If r ≤ a: Area = (1/4)πr² (quarter circle at corner)
- If r > a: More complex — subtract the area not accessible
CBSE Exam Pattern: Combination area problems are 3-4 marks. Standard: find shaded area = whole - holes, or sum of parts.
Perimeters and Mixed Applications
Perimeter of Combined Figures
Perimeter = Sum of all boundary segments (straight lines + arcs).
Key: Do NOT include internal lines in perimeter!
Examples of Perimeters:
| Figure | Perimeter |
|---|---|
| Semicircle | πr + 2r |
| Sector (angle θ) | 2r + arc = 2r + (θ/360°)×2πr |
| Sector-based petal | 2 × arc length |
Approach for Standard Problems
Step 1: Identify all boundary curves (arcs, straight lines) Step 2: Calculate each piece separately Step 3: Add all pieces for total perimeter
Special Cases — Quick Results
| Shape | Area | Perimeter |
|---|---|---|
| Circle radius r | πr² | 2πr |
| Semicircle | πr²/2 | πr+2r |
| Ring (outer R, inner r) | π(R²-r²) | 2πR+2πr |
| Sector angle θ | θπr²/360° | 2r+θ(2πr)/360° |
Area of Circular Path
A circular path of width w surrounds a circle of radius r:
- Outer radius = r + w
- Area of path = π(r+w)² - πr² = π(2rw + w²) = πw(2r+w)
HBSE Tip: Questions about 'area of path' and 'sector-segment' are direct formula applications. Memorize formulas and substitute carefully.
Frequently asked questions
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Yes — the Areas Related to Circles notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Areas Related to Circles notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Areas Related to Circles chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Areas Related to Circles.