TEACHGYAN • CLASS 10 MATHEMATICS • NCERT 2026–27

Chapter 12: Surface Areas and Volumes

Detailed Notes • Worked Examples • Competency • Board Practice

📖 Chapter Explanation

Three-dimensional geometry extends measurement from flat figures to solids. Students calculate curved and total surface areas and volumes of cylinders, cones, spheres and combinations of solids, while keeping units consistent.

🧭 Concept Journey

📌 Core Concepts

For a cylinder, curved surface area=2πrh and volume=πr²h. For a sphere, surface area=4πr² and volume=4/3πr³. For a cone, slant height l=√(r²+h²), curved surface area=πrl and volume=1/3πr²h.

💡 Worked Example

🔍 Why the Method Works

For a cylinder, curved surface area=2πrh and volume=πr²h. For a sphere, surface area=4πr² and volume=4/3πr³. For a cone, slant height l=√(r²+h²), curved surface area=πrl and volume=1/3πr²h.

⚠️ Common Errors

Do not mix radius and diameter. For a cone, slant height is not the vertical height. Convert units before combining measurements.

🧠 Concept-Based MCQs

1. Volume of a cylinder is: A) 2πrh B) πr²h C) πrl D) 4πr²
2. The slant height of a cone is: A) r+h B) √(r²+h²) C) r²+h² D) h−r
3. Volume is measured in: A) cm B) cm² C) cm³ D) degrees
4. A sphere’s surface area is: A) 2πr² B) 3πr² C) 4πr² D) 4πr³

Answer Key: 1-B, 2-B, 3-C, 4-C.

📊 Case-Based / Competency

🔎 Extension Case

A storage container combines a cylinder and hemisphere. Students calculate its volume, identify the surfaces requiring paint and explain which internal boundaries should not be counted.

🔥 HOTS & Board Challenge

1. Explain why the selected formula or method is appropriate.
2. Create a similar problem and solve it.
3. Find and correct a plausible incorrect solution.
4. Apply the concept in a new context.

✍️ Exam Practice

1. Define the central concept.
2. Solve a direct application problem.
3. Solve a multi-step competency problem.
4. Interpret a diagram/table/context.
5. Write a complete 5-mark solution with reasoning.

📄 Practice Worksheet

1. 5 concept checks
2. 5 standard problems
3. 3 applications
4. 2 case-based questions
5. 2 HOTS questions
6. 1 board-style challenge

🎯 One-Minute Revision

solid → dimensions → net/curved surface → total surface → volume → composite solid → unit conversion

🏆 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.