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I. Chapter Summary
This chapter introduces students to the surface areas and volumes of three-dimensional solids, including cubes, cuboids, cylinders, cones, spheres, and hemispheres. It focuses on deriving and applying standard formulas to calculate the curved surface area (CSA), total surface area (TSA), and volume of these solids. The chapter emphasizes real-world applications and includes problems involving combinations of solids.
II. Key Concepts Covered
| Concept | Explanation |
| Surface Area | The total area that the surface of a solid object occupies. |
| Volume | The amount of space occupied by a three-dimensional object. |
| Cube and Cuboid |
- TSA of cube = 6a²
- TSA of cuboid = 2(lb + bh + hl)
- Volume of cube = a³
- Volume of cuboid = l × b × h |
| Cylinder | - CSA = 2πrh
- TSA = 2πr(h + r)
- Volume = πr²h |
| Cone | - Slant height (l) = √(r² + h²)
- CSA = πrl
- TSA = πr(l + r)
- Volume = (1/3)πr²h |
| Sphere and Hemisphere | - Sphere:
• CSA = 4πr²
• Volume = (4/3)πr³ - Hemisphere:
• CSA = 3πr²
• Volume = (2/3)πr³ |
III. Important Questions
(A) Multiple Choice Questions (1 Mark)
- The volume of a cube of side 4 cm is:
a) 64 cm³ ✔️
b) 16 cm³
c) 48 cm³
d) 32 cm³ - CSA of a cylinder =
a) 2πrh ✔️
b) 2πr²
c) πr²h
d) πrl - The volume of a cone is given by:
a) πr²h
b) (1/2)πr²h
c) (1/3)πr²h ✔️
d) (2/3)πr²h - TSA of a hemisphere is:
a) 2πr²
b) 3πr² ✔️
c) 4πr²
d) πr²
(B) Short Answer Questions (2/3 Marks)
- A cube has a side of 5 cm. Find its surface area.
- A cylinder has a radius of 3.5 cm and height 10 cm. Find its volume. (PYQ 2019)
- Find the CSA of a cone with radius 7 cm and slant height 25 cm.
- A cuboid has dimensions 4 cm × 5 cm × 6 cm. Find its TSA.
(C) Long Answer Questions (5 Marks)
- Find the volume and surface area of a sphere with radius 10.5 cm. (PYQ 2018)
- A hemispherical bowl has radius 7 cm. Find its CSA and volume.
- A tent is in the shape of a right circular cone. The radius of the base is 10.5 m and the height is 24 m. Find the CSA and volume of the tent. (PYQ 2020)
- A cubical tank has a capacity of 729000 cm³. Find the length of its edge and TSA.
(D) HOTS (Higher Order Thinking Skills)
- A container is in the form of a cylinder with hemispherical ends. Find the total surface area and volume of the container.
- How many spherical balls of radius 2 cm can be made by melting a metal cylinder of radius 8 cm and height 15 cm? Justify with calculations.
IV. Key Formulas/Concepts
| Solid | Surface Area | Volume |
| Cube | TSA = 6a² | a³ |
| Cuboid | TSA = 2(lb + bh + hl) | l × b × h |
| Cylinder | CSA = 2πrh, TSA = 2πr(h + r) | πr²h |
| Cone | CSA = πrl, TSA = πr(l + r) | (1/3)πr²h |
| Sphere | TSA = 4πr² | (4/3)πr³ |
| Hemisphere | CSA = 2πr², TSA = 3πr² | (2/3)πr³ |
Note:
- π = 3.14 or (22/7)
- Slant height of cone: l = √(r² + h²)
V. Deleted Portions (CBSE 2025–2026)
No portions have been deleted from this chapter as per the rationalized NCERT textbooks.
VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026)
| Chapter | Estimated Marks | Type of Questions Typically Asked |
| Surface Areas and Volumes | 7–10 Marks | 1 MCQ, 1 Short Calculation, 2 Application-Based Long Questions |
VII. Previous Year Questions (PYQs)
| Marks | Question | Year |
| 3 marks | Find the volume of a cone of radius 7 cm and height 12 cm. | PYQ 2020 |
| 3 marks | CSA and volume of a hemisphere of radius 14 cm. | PYQ 2019 |
| 5 marks | A tent is a cone of radius 10.5 m and height 24 m. Find its CSA and volume. | PYQ 2018 |
| 2 marks | Volume of a cuboid tank with known dimensions. | PYQ 2018 |
VIII. Real-World Application Examples
- Architecture & Construction: Tanks, domes, pillars, and storage spaces.
- Packaging & Manufacturing: Cylindrical containers, cone-shaped cups, spherical balls.
- Engineering: Design of industrial machines, pipelines, silos.
- Everyday Life: Measuring paint for walls, wrapping gifts, determining water capacity.
IX. Student Tips & Strategies for Success
Time Management
- Day 1: Memorize all formulas with meaning and diagrams
- Day 2: Practice direct formula-based problems
- Day 3: Solve real-life application and HOTS problems
Exam Preparation
- Keep a formula sheet and revise daily
- Label all units properly (cm², m³, etc.)
- Show step-by-step calculations and conversions
Stress Management
- Visualize solids using models or real objects
- Practice questions with increasing difficulty level
X. Career Guidance & Exploration
- For Classes 9–10:
➤ Builds foundational geometry for Olympiads and NTSE
➤ Helps in daily problem solving, design thinking - For Classes 11–12 & Careers:
➤ Useful in:
• Architecture
• Civil & Mechanical Engineering
• Industrial Design
• Packaging & Logistics - Exams Where Relevant:
➤ JEE (Volume and 3D geometry problems)
➤ CUET
➤ NID/UCEED (Design aptitude with geometry)
➤ NDA (Mensuration-based reasoning)
XI. Important Notes
Always include units and π value (approx.) in all answers.
Diagrams help visualize problems—draw them even if not asked.
Solve mixed problems with different solids to develop better understanding.
Round off answers only if instructed in the question.