🍕 Have You Ever Wondered?
When you buy an 8-inch pizza slice or watch a car's windshield wiper sweep across the glass, how do you calculate the exact surface area cleaned? Sect...
When you buy an 8-inch pizza slice or watch a car's windshield wiper sweep across the glass, how do you calculate the exact surface area cleaned? Sectors and segments apply circular fractions to solve real-world areas.
Why This Chapter Matters
In Class 10 Mathematics, "Areas Related to Circles" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
Before You Begin (Prerequisites)
- Circumference ($2\pi r$) and area of a circle ($\pi r^2$).
- Angles and degrees ($360^\circ$).
- Area of triangles.
What You Will Learn (Core Objectives)
- Calculate the perimeter (circumference) and area of a circle.
- Calculate the Area of a Sector of angle $\theta$: $\frac{\theta}{360^\circ}\pi r^2$.
- Calculate the Length of an Arc of a sector: $\frac{\theta}{360^\circ}2\pi r$.
- Calculate the Area of a Major Sector and Minor/Major Segments of a circle.
- Solve practical problems involving windshield wipers, clock hands, and brooch designs.
Chapter Roadmap & Progression
1
1. Area of a Sector of a Circle
2
2. Area of a Segment of a Circle
3
3. Clock Hands and Angular Speed
Complete Concept Guide (100% Curriculum Coverage)
1. Area of a Sector of a Circle
A Sector is the pie-slice region enclosed by two radii and the corresponding arc: $$\mathbf{\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2} \quad \text{and} \quad \mathbf{\text{Length of Arc } l = \frac{\theta}{360^\circ} \times 2\pi r}$$
2. Area of a Segment of a Circle
A Segment is the region bounded by a chord and its corresponding arc: $$\mathbf{\text{Area of Minor Segment} = \text{Area of Sector } OAB - \text{Area of } \triangle OAB}$$
3. Clock Hands and Angular Speed
A clock face has $360^\circ$ divided into 60 minutes:
• Minute hand sweeps $\frac{360^\circ}{60} = \mathbf{6^\circ\text{ per minute}}$!
• In 5 minutes, it sweeps $5 \times 6^\circ = 30^\circ$.
Visual Learning & Conceptual Map
Areas Related to Circles Master Matrix
Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture
1. Area of a Sector of a Circle • 2. Area of a Segment of a Circle
Chapter Summary & 10 Key Takeaways
Takeaway 1
Sector Area: $\frac{\theta}{360}\pi r^2$.
Takeaway 2
Arc Length: $\frac{\theta}{360}2\pi r$.
Takeaway 3
Segment Area: $\text{Sector Area} - \text{Triangle Area}$.
Takeaway 4
Minute Hand Rate: Sweeps $6^\circ$ every minute.
Takeaway 5
Major Sector Area: $\pi r^2 - \text{Minor Sector Area}$.
Check Your Understanding (Diagnostic Practice Questions)
Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.
1
Find the area of a sector of a circle of radius 6 cm if angle of sector is $60^\circ$.
Reveal Answer & Explanation
Answer: $\text{Area} = \frac{60}{360} \times \frac{22}{7} \times 6^2 = \frac{1}{6} \times \frac{22}{7} \times 36 = \frac{132}{7} = 18.86\text{ cm}^2$.
Area = 132/7 cm^2.
2
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
Reveal Answer & Explanation
Answer: In 5 min, angle $\theta = 5 \times 6^\circ = 30^\circ$. $\text{Area} = \frac{30}{360} \times \frac{22}{7} \times 14^2 = \frac{1}{12} \times \frac{22}{7} \times 196 = \frac{154}{3} = 51.33\text{ cm}^2$.
Area = 154/3 cm^2.
3
A chord of a circle of radius 10 cm subtends a right angle at the center. Find the area of the minor segment (use $\pi = 3.14$).
Reveal Answer & Explanation
Answer: $\text{Sector Area} = \frac{90}{360}(3.14)(10^2) = 78.5\text{ cm}^2$. $\text{Triangle Area} = \frac{1}{2}(10)(10) = 50\text{ cm}^2$. $\text{Segment Area} = 78.5 - 50 = 28.5\text{ cm}^2$.
Segment area = 28.5 cm^2.
4
An umbrella has 8 ribs which are equally spaced. Assuming umbrella to be a flat circle of radius 45 cm, find the area between two consecutive ribs.
Reveal Answer & Explanation
Answer: Each sector angle $\theta = \frac{360^\circ}{8} = 45^\circ$. $\text{Area} = \frac{1}{8}\pi r^2 = \frac{1}{8} \times \frac{22}{7} \times 45^2 = \frac{22275}{28}\text{ cm}^2 \approx 795.54\text{ cm}^2$.
Area = 22275/28 cm^2.
5
What is the relationship between the area of a sector, its arc length $l$, and radius $r$?
Reveal Answer & Explanation
Answer: $\text{Area of Sector} = \frac{1}{2} \times l \times r$.
Area = 1/2 * l * r.
Finished Studying This Chapter?
READY TO PRACTICE?
Timed CBT Practice Tests (Exam Simulator)
Put your concepts to the test with official curriculum-aligned Foundation and Advanced practice tests. Get instant accuracy scores, time metrics, and step-by-step verified explanations.
