CBSE Class 10 Maths Notes - Chapter 12 Areas Related to Circles
Introduction
Welcome to these comprehensive CBSE Class 10 Notes for Mathematics Chapter 12: Areas Related to Circles. This chapter builds upon your previous knowledge of circles from Class 9 and introduces you to the calculation of areas and lengths associated with circular sectors and segments.
Why this chapter matters:
- High Scoring: Areas Related to Circles carries approximately 3-5 marks in the CBSE Class 10 Board Exam
- Foundation Building: Essential for Class 11-12 Geometry and Trigonometry
- Real-world Applications: Used in engineering, architecture, and design
These CBSE Class 10 Maths Notes are 100% aligned with the latest NCERT syllabus and are perfect for board exam preparation, school exams, and competitive foundation courses.
Chapter Overview
These CBSE Class 10 Notes cover the following key topics:
- Circumference of Circle - Complete boundary measurement
- Area of Circle - Region enclosed by the circle
- Area of Sector - Pie-slice portion of circle
- Length of Arc - Curved portion of sector boundary
- Area of Segment - Region between chord and arc
- Word Problems - Real-life application questions
Key Concepts (NCERT Accurate)
Circumference of Circle
The distance around the circle is called its circumference.
- Where r = radius of the circle
- Ï€ (pi) ≈ 22/7 or 3.14
Area of Circle
The region occupied by the circle in a 2D plane.
Area of Sector
A sector is the region enclosed by two radii and an arc.
- Where θ = central angle in degrees
- r = radius of circle
Derivation: Since the complete angle at center is 360°, the sector represents θ/360° of the total circle area.
Length of Arc
The curved part of the sector boundary.
Area of Segment
A segment is the region between a chord and its corresponding arc.
- Area of Triangle = 1/2 × r² × sin θ (when two sides are radii)
- Or use standard triangle area formula if base and height are known
Important Formulas (Quick Reference)
| Formula | Expression | When to Use |
|---|---|---|
| Circumference | C = 2Ï€r | Finding boundary length |
| Area of Circle | A = Ï€r² | Total area enclosed |
| Area of Sector | (θ/360°) × Ï€r² | Pie-slice area calculation |
| Length of Arc | (θ/360°) × 2Ï€r | Curved boundary length |
| Area of Segment | Sector Area - Triangle Area | Region between chord and arc |
Solved Examples (CBSE Pattern)
Example 1: Area of Sector (Easy)
Question: Find the area of a sector of a circle with radius 6 cm and angle 60°. (Use Ï€ = 22/7)
Solution:
Given: r = 6 cm, θ = 60°, Ï€ = 22/7
Formula: Area of Sector = (θ/360°) × Ï€r²
= (60/360) × (22/7) × 6 × 6
= (1/6) × (22/7) × 36
= (22 × 6)/7
= 132/7 cm² or 18.86 cm²
Example 2: Arc Length (Moderate)
Question: Find the length of an arc of a circle with radius 21 cm subtending an angle of 120° at the center.
Solution:
Given: r = 21 cm, θ = 120°
Formula: Arc Length = (θ/360°) × 2Ï€r
= (120/360) × 2 × (22/7) × 21
= (1/3) × 2 × 22 × 3
= 44 cm
Example 3: Area of Segment
Question: Find the area of the segment of a circle with radius 14 cm and central angle 60°. (Use Ï€ = 22/7, √3 = 1.732)
Solution:
Given: r = 14 cm, θ = 60°
Step 1: Area of Sector = (60/360) × (22/7) × 14 × 14
= (1/6) × 22 × 2 × 14
= 616/6 = 102.67 cm²
Step 2: Area of Triangle = 1/2 × r² × sin θ
= 1/2 × 14 × 14 × sin 60°
= 1/2 × 196 × (√3/2)
= 98 × 0.866 = 84.87 cm²
Step 3: Area of Segment = 102.67 - 84.87
= 17.80 cm²
Example 4: Board-Level Word Problem (5 Marks)
Question: A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades. (Use Ï€ = 22/7)
Solution:
Given: r = 25 cm, θ = 115°, Number of wipers = 2
Formula: Area of Sector = (θ/360°) × Ï€r²
Area cleaned by one wiper:
= (115/360) × (22/7) × 25 × 25
= (115/360) × (22/7) × 625
= (115 × 22 × 625)/(360 × 7)
= 1,581,250/2,520
= 627.48 cm²
Total area cleaned by 2 wipers:
= 2 × 627.48
= 1254.96 cm² ≈ 1255 cm²
Smart Tricks for Board Exams
Ï€ Value Selection Trick
- Use 22/7 when radius is divisible by 7 (e.g., 7, 14, 21, 28)
- Use 3.14 when radius is not divisible by 7
- Exam Tip: Check the question - it usually specifies which to use
Degree Fraction Simplification
- 30° = 1/12, 45° = 1/8, 60° = 1/6, 90° = 1/4, 120° = 1/3, 180° = 1/2
- Simplify before multiplying to save calculation time
Segment Shortcut
- If θ = 60°, triangle is equilateral (all sides = r)
- If θ = 90°, use 1/2 × r² for triangle area
- If θ = 120°, use properties of 30-60-90 triangles
Unit Conversion Reminder
- Always convert diameter to radius (r = d/2) before applying formulas
- Maintain consistent units throughout (cm → cm², m → m²)
Time-Saving Strategy
- Write the formula first (carries 1/2 mark)
- Show clear substitution step
- Box your final answer with units
Visual Learning
Figure 1: Circle with radius and central angle
Figure 2: Area of sector derivation
Figure 3: Arc length representation
Figure 4: Major and Minor segments
Most Important Board Questions
1 Mark Questions
- What is the formula for the area of a sector? (1M)
- Define a segment of a circle. (1M)
- Find the circumference of a circle with radius 7 cm. (1M)
2-3 Mark Questions
- Find the area of a quadrant of a circle whose circumference is 22 cm. (2M)
- * The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. (3M)
- * A chord of a circle of radius 10 cm subtends a right angle at the center. Find the area of the corresponding minor segment. (3M)
4-5 Mark Word Problems
- ** In a circle of radius 21 cm, an arc subtends an angle of 60° at the center. Find:
- (i) Length of the arc
- (ii) Area of the sector
- (iii) Area of the segment
- * A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors. Find:
- (i) Total length of silver wire required
- (ii) Area of each sector of the brooch
Case-Study Based Question
- ** Case Study: An umbrella has 8 ribs which are equally spaced. Assuming the umbrella to be a flat circle of radius 45 cm, find the area between two consecutive ribs of the umbrella. (4M)
- Area of sector problems with clock hands
- Segment area calculations
- Combination figures (circle + triangle)
- Wheel rotation problems
Common Mistakes to Avoid
| Mistake | Correction |
|---|---|
| Using diameter instead of radius | Always use r; convert d to r (r = d/2) |
| Forgetting to convert degrees | Ensure θ is in degrees for the formula |
| Missing units in final answer | Always write cm², m², cm, etc. |
| Using wrong π value | Use 22/7 when r is multiple of 7 |
| Confusing sector with segment | Sector = pie slice; Segment = chord + arc region |
| Calculation errors in fractions | Simplify θ/360° first (e.g., 60/360 = 1/6) |
Practice Section
MCQs
Q1. If the circumference of a circle is 44 cm, then its area is:
- (a) 154 cm²
- (b) 308 cm²
- (c) 77 cm²
- (d) 44 cm²
Q2. The area of a sector of angle θ (in degrees) of a circle with radius R is:
- (a) (θ/180) × 2Ï€R
- (b) (θ/360) × Ï€R²
- (c) (θ/360) × 2Ï€R
- (d) (θ/180) × Ï€R²
Assertion-Reason
Q3.
Assertion (A): The area of a circle is 154 cm², then its perimeter is 44 cm.
Reason (R): If the area of a circle is A, then its perimeter is √(4Ï€A).
Choose the correct option:
- (a) Both A and R are true and R is the correct explanation of A
- (b) Both A and R are true but R is not the correct explanation of A
- (c) A is true but R is false
- (d) A is false but R is true
Case-Study Based
Q4. A racing track is in the form of a ring with inner circumference 352 m and outer circumference 396 m. Find:
- (i) Width of the track
- (ii) Area of the track (Use π = 22/7)
HOTS Questions
Q5. A round table cover has six equal designs as shown in the figure. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs.0.35 per cm². (Use √3 = 1.7)
Frequently Asked Questions (FAQ)
Is Areas Related to Circles important for CBSE Class 10 board exam?
Yes, absolutely! This chapter carries 3-5 marks in the board exam and is considered one of the easier, high-scoring chapters in CBSE Class 10 Maths Notes. Questions are predictable and formula-based.
When should I use 22/7 and when to use 3.14?
Use 22/7 when the radius or diameter is a multiple of 7 (like 7, 14, 21, 28, 35). Use 3.14 for other values. The question usually specifies which value to use. These CBSE Class 10 Notes recommend checking the question paper instructions first.
Are these CBSE Class 10 Maths Notes based on NCERT?
Yes, 100%! These CBSE Class 10 Notes are completely aligned with the latest NCERT textbook for Class 10 Mathematics. All formulas, examples, and concepts follow the NCERT pattern and are perfect for board exam preparation.
How can I download these Class 10 Maths Notes PDF?
These notes are designed for easy copying or conversion to PDF format. Save this page or copy the content to create your own Class 10 Maths Notes PDF for offline revision.
Conclusion
Mastering Areas Related to Circles requires thorough understanding of formulas and consistent practice. These CBSE Class 10 Maths Notes provide you with everything needed to score full marks in this chapter - from basic concepts to advanced problem-solving techniques.
- Memorize all formulas from the boxed section
- Practice the solved examples without looking at solutions
- Solve previous year board questions (last 5 years)
- Time yourself while solving to improve speed
Next Chapter: Continue your preparation with Surface Areas and Volumes - CBSE Class 10 Notes for comprehensive coverage of 3D geometry concepts.
Keywords: CBSE Class 10 Notes, CBSE Class 10 Maths Notes, Class 10 Maths Chapter 12 Notes, Areas Related to Circles Class 10, Sector and Segment Class 10, NCERT Class 10 Maths Notes, Class 10 Maths Notes PDF
