TEACHGYAN • CLASS 10 MATHEMATICS • NCERT 2026–27
Chapter 11: Areas Related to Circles
Detailed Notes • Worked Examples • Competency • Board Practice
📖 Chapter Explanation
This chapter applies circle geometry to calculate lengths and regions involving circles, sectors and segments. The emphasis is on interpreting a diagram correctly before selecting the appropriate formula.
🧭 Concept Journey
📌 Core Concepts
Circumference of a circle is 2πr and area is πr². For a sector of angle θ°, arc length = θ/360 × 2πr and sector area = θ/360 × πr².
💡 Worked Example
A 90° sector of radius 14 cm has arc length 1/4 × 2π×14 = 7π cm and area 1/4 × π×14² = 49π cm².
🔍 Why the Method Works
Circumference of a circle is 2πr and area is πr². For a sector of angle θ°, arc length = θ/360 × 2πr and sector area = θ/360 × πr².
⚠️ Common Errors
The most frequent error is using the full-circle formula for a sector without multiplying by θ/360. Also distinguish arc length from sector area.
🧠 Concept-Based MCQs
- The circumference of radius r is: A) πr² B) 2πr C) πd² D) r/2
- A 90° sector is what fraction of a circle? A) 1/2 B) 1/3 C) 1/4 D) 1/6
- Sector area depends on: A) only diameter B) radius and central angle C) only circumference D) only arc length
- Arc length is measured in: A) square units B) cubic units C) linear units D) degrees
Answer Key: 1-B, 2-C, 3-B, 4-C.
📊 Case-Based / Competency
A circular garden contains a 60° flower bed. Given its radius, calculate the curved boundary and planted area and explain why both require different formulas.
- Identify the information and variables.
- Select the appropriate mathematical model.
- Calculate step by step.
- Interpret and verify the result.
🔎 Extension Case
A circular park has a sector reserved for a play area. Students calculate the sector’s area, remaining area and boundary length, then explain the difference between each measurement.
🔥 HOTS & Board Challenge
- Explain why the selected formula or method is appropriate.
- Create a similar problem and solve it.
- Find and correct a plausible incorrect solution.
- Apply the concept in a new context.
✍️ Exam Practice
- Define the central concept.
- Solve a direct application problem.
- Solve a multi-step competency problem.
- Interpret a diagram/table/context.
- Write a complete 5-mark solution with reasoning.
📄 Practice Worksheet
- 5 concept checks
- 5 standard problems
- 3 applications
- 2 case-based questions
- 2 HOTS questions
- 1 board-style challenge
🎯 One-Minute Revision
circle → radius/diameter → circumference → sector → arc length → area of sector → segment → composite figures
🏆 Mastery Checklist
- ☐ I understand the concept.
- ☐ I can select the correct method.
- ☐ I can solve step by step.
- ☐ I can handle competency questions.
- ☐ I can verify and interpret my answer.
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Original explanatory study material aligned to NCERT/CBSE topics; textbook passages are not reproduced verbatim.