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I. Chapter Summary:
This chapter focuses on the surface area and volume calculations of 3D geometrical figures such as cubes, cuboids, cylinders, cones, spheres, hemispheres, and their combinations. Students learn how to apply these formulas in real-world problems involving capacity, cost estimation, material usage, and practical designs. It also includes conversion of solids from one shape to another while conserving volume.
II. Key Concepts Covered:
| Concept | Description |
| Surface Area | The total area covering the outer surface of a 3D object. |
| Volume | The space occupied by a 3D object. |
| Curved Surface Area (CSA) | Area of the curved portion only (e.g., side of a cylinder). |
| Total Surface Area (TSA) | Sum of CSA and area of flat surfaces (like top and base). |
| Combination of Solids | Finding surface area or volume of shapes formed by combining two or more solids. |
| Conversion of Solids | When one solid is melted and recast into another, volume remains constant. |
III. Important Questions:
(A) Multiple Choice Questions (1 Mark):
- The volume of a cylinder is given by:
a) $\pi r^2 h$✅
b) $2\pi r h$
c) $\pi r h^2$
d) $\pi r^2 + h$ - The total surface area of a sphere is:
a) $2\pi r^2$
b) $\pi r^2$
c) $4\pi r^2$ ✅
d) $3\pi r^2$ - A cube has side 5 cm. What is its surface area?
a) 25 cm²
b) 150 cm²
c) 125 cm²
d) 150 cm² ✅ - If a cone is melted and recast into a cylinder of same base radius, the height of the cylinder will be:
a) Equal to cone’s height
b) One-third of cone’s height
c) One-third of cone’s height ✅
d) Three times cone’s height
(B) Short Answer Questions (2/3 Marks):
- A cylindrical water tank has a radius of 1.5 m and height 2 m. Find its total surface area.
(PYQ 2020) - Find the volume of a cone with base radius 7 cm and height 24 cm.
- A spherical ball is melted to form 8 smaller balls of equal size. Find the radius of each if the original radius was 6 cm.
- A hemisphere bowl has radius 7 cm. Find its curved surface area.
(C) Long Answer Questions (5 Marks):
- A solid metallic cone of radius 6 cm and height 8 cm is melted and recast into spheres of radius 1 cm each. How many such spheres will be formed?
(PYQ 2021) - A cylindrical pillar is 50 cm in diameter and 3.5 m high. Find the cost of painting the curved surface at ₹12 per m².
- A toy is in the shape of a cone mounted on a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 4 cm. Find the total surface area of the toy.
- A cylindrical container with base radius 7 cm and height 15 cm is filled with water. How many spherical balls of radius 1.5 cm can be submerged in it without overflowing?
(D) HOTS (Higher Order Thinking Skills):
- A rectangular solid block is cut into two right circular cones of equal volume. Prove that the total surface area changes and find the % increase or decrease in surface area.
- A cone is surmounted on a hemisphere. The height of the cone is equal to the radius of the hemisphere. Prove that the total surface area of the solid is $5\pi r^2$.
IV. Key Formulas/Concepts:
| Solid | Curved Surface Area (CSA) | Total Surface Area (TSA) | Volume |
| Cuboid | $2h(l + b)$ | $2(lb + bh + hl)$ | $l \times b \times $h$ |
| Cube | $4a^2$ | $6a^2$ | $a^3$ |
| Cylinder | $2\pi r h$ | $2\pi r(h + r)$ | $\pi r^2 h$ |
| Cone | $\pi r l$ | $\pi r(l + r)$ | $\frac{1}{3} \pi r^2 h$ |
| Sphere | – | $4\pi r^2$ | $\frac{4}{3} \pi r^3$ |
| Hemisphere | $2\pi r^2$ | $3\pi r^2$ | $\frac{2}{3} \pi r^3$ |
| Frustum | $\pi (R + r) l$ | $\pi(R + r)l + \pi R^2 + \pi r^2$ | $\frac{1}{3} \pi h (R^2 + r^2 + Rr)$ |
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 2025–26):
| Unit/Chapter | Estimated Marks | Type of Questions Typically Asked |
| Surface Areas and Volumes | 6–8 Marks | 1 Short Answer + 1 Long Answer or HOTS |
VII. Previous Year Questions (PYQs):
| Year | Marks | Question |
| 2019 | 5M | Cone melted into spheres – calculate number of spheres |
| 2020 | 3M | TSA of a cylindrical tank |
| 2021 | 5M | Recasting of solids into smaller units |
| 2022 | 3M | TSA/Volume-based application question on toys or tanks |
VIII. Real-World Application Examples:
| Real-Life Scenario | Concept Used |
| Designing cylindrical water tanks | Volume of cylinders |
| Manufacturing spherical balls | Volume and surface area of spheres |
| Estimating material for painting pipes or tanks | Curved surface area |
| Melting and reshaping metals | Volume conservation in recasting |
IX. Student Tips & Strategies for Success:
Time Management:
- Allocate 30 mins every alternate day for problem-solving from this chapter.
- Use flowcharts to memorize formulas.
Exam Preparation:
- Create a formula sheet for each shape.
- Focus on unit conversions (cm to m, m³ to liters, etc.).
- Practice questions involving combinations of solids.
Stress Management:
- Practice with visual aids or 3D models to better understand concepts.
- Use past papers to simulate exam conditions.
X. Career Guidance & Exploration (Class-Specific):
For Classes 9–10:
| Stream | Career Paths | Related Exams |
| Science | Civil Engineer, Architect, Product Designer | NTSE, Olympiads |
| Commerce | Statistician, Logistics Manager | Maths Olympiads |
| Arts | Fashion/Interior Designer | NID/UCEED (Geometry use) |
Tip: These concepts are foundational for competitive exams like JEE (Mensuration), NDA, CUET, and practical fields like architecture and manufacturing.
XI. Important Notes:
- Refer to https://cbseacademic.nic.in and https://ncert.nic.in for updates.
- Always write formulas before solving.
- Maintain a concept + formula notebook.
- Regular practice builds confidence and accuracy.