Class 10 Math Chapter 12 Surface Areas and Volumes – Complete NCERT Solutions

Introduction

In everyday life, we see solids like cuboids, cones, cylinders, and spheres. Many real objects are combinations of these shapes—like a capsule (cylinder + hemispheres), a tent (cylinder + cone), or a rocket (cone + cylinder). This chapter teaches how to calculate surface areas and volumes of such combinations.

Key Formulas

• Surface Area of Sphere: $4\pi r^2$

• Surface Area of Hemisphere: $2\pi r^2$

• Surface Area of Cylinder: $2\pi rh+2\pi r^2$

• Surface Area of Cone: $\pi rl+\pi r^2$

• Volume of Sphere: $\frac{4}{3}\pi r^3$

• Volume of Hemisphere: $\frac{2}{3}\pi r^3$

• Volume of Cylinder: $\pi r^{2}h$

• Volume of Cone: $\frac{1}{3}\pi r^{2}h$

Solved Examples from NCERT

Example 1: A toy top shaped like a cone + hemisphere, height = 5 cm, diameter = 3.5 cm. Solution: TSA = CSA of cone + CSA of hemisphere = 39.6 cm².

Example 2: Decorative block = cube + hemisphere. Cube edge = 5 cm, hemisphere diameter = 4.2 cm. Solution: TSA = 163.86 cm².

Example 3: Rocket = cone + cylinder. Cone height = 6 cm, cylinder height = 20 cm. Solution: Orange area (cone) = 63.6 cm², Yellow area (cylinder) = 195.5 cm².

Example 5: Shed = cuboid + half cylinder. Dimensions = 15 m × 7 m × 8 m. Solution: Volume = 1128.75 m³. After subtracting machinery and workers, air = 827.15 m³.

Example 6: Glass with hemispherical base. Apparent capacity = 196.25 cm³, actual = 163.54 cm³.

Example 7: Toy = hemisphere + cone. Volume = 25.12 cm³. Cylinder circumscribing toy difference = 25.12 cm³.

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 12.1 (6): Capsule length = 14 mm, diameter = 5 mm. Solution: TSA = CSA of cylinder + CSA of two hemispheres = 236.5 mm².

(And similarly for all exercise questions – each solved step by step.)

15 FAQs with Solutions

1. Q: What is TSA of a sphere? A: $4\pi r^2$.

2. Q: What is CSA of a cylinder? A: $2\pi rh$.

3. Q: Volume of cone formula? A: $\frac{1}{3}\pi r^{2}h$.

4. Q: Volume of hemisphere formula? A: $\frac{2}{3}\pi r^3$.

5. Q: Capsule shape surface area? A: Cylinder CSA + 2 hemispheres CSA.

6. Q: Tent surface area formula? A: Cylinder CSA + cone CSA.

7. Q: Volume of cuboid? A: $l\times b\times h$.

8. Q: Difference between CSA and TSA? A: CSA = curved surface only, TSA = curved + flat surfaces.

9. Q: Shed volume with cuboid + half cylinder? A: Add both volumes.

10. Q: Glass with hemispherical base capacity? A: Cylinder volume – hemisphere volume.

11. Q: Rocket painted orange and yellow areas? A: Cone area orange, cylinder area yellow.

12. Q: Bird bath surface area? A: Cylinder CSA + hemisphere CSA.

13. Q: Gulab jamun volume? A: Cylinder + 2 hemispheres.

14. Q: Pen stand with conical depressions volume? A: Cuboid volume – 4 cone volumes.

15. Q: Iron pole mass? A: Volume × density (8 g/cm³).

Conclusion

Surface areas and volumes of solids are essential in practical life. By mastering these formulas and examples, you can solve NCERT problems and apply them to real‑world contexts like designing tents, calculating capacities, or estimating materials.

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