Class 9 Maths Chapter 11 Surface Areas and Volumes – Exercise 11.6 NCERT Solutions

Introduction

Exercise 11.6 focuses on surface areas and volumes of combinations of solids. Students learn how to calculate the surface area and volume of objects formed by combining two or more basic solids such as cylinders, cones, hemispheres, and spheres. This exercise is important for solving real‑life mensuration problems.

Key Concepts

1. Combination of Solids:

   • Add volumes of individual solids to get total volume.

   • Add curved surface areas and bases as required to get total surface area.

2. Common Combinations:

   • Cylinder + Hemisphere (e.g., water tanks).

   • Cone + Hemisphere (e.g., ice‑cream cones).

   • Cylinder + Cone (e.g., tent structures).

Common Mistakes

• Forgetting to exclude hidden surfaces when calculating TSA.

• Using diameter instead of radius.

• Confusing CSA with TSA.

• Not converting units consistently.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. A toy is in the form of a hemisphere surmounted by a cone. The height of cone is 2 cm and radius is 3.5 cm. Find volume of toy.

$$V=\frac{1}{3}\pi r^{2}h+\frac{2}{3}\pi r^3$$

$$=\frac{1}{3}\times \frac{22}{7}\times (3.5{)}^2\times 2+\frac{2}{3}\times \frac{22}{7}\times (3.5{)}^3$$

$$=25.67+89.6=115.27{\text{cm}}^3$$

Q2. A capsule is in the form of a cylinder with hemispherical ends. The length of capsule is 14 cm and radius is 2 cm. Find volume. Cylinder height = 14 – 2×2 = 10 cm.

$$V=\pi r^{2}h+\frac{4}{3}\pi r^3$$

$$=\frac{22}{7}\times 4\times 10+\frac{4}{3}\times \frac{22}{7}\times 8$$

$$=125.7+33.5=159.2{\text{cm}}^3$$

Q3. A circus tent is cylindrical up to height 3 m and conical above with height 4 m. Radius = 2.1 m. Find volume.

$$V=\pi r^2{h}_{cyl}+\frac{1}{3}\pi r^2{h}_{cone}$$

$$=\frac{22}{7}\times (2.1{)}^2\times 3+\frac{1}{3}\times \frac{22}{7}\times (2.1{)}^2\times 4$$

$$=41.58+26.39=67.97{\text{m}}^3$$

Q4. A toy is in the form of a cone surmounted on a hemisphere. Radius = 7 cm, height of cone = 24 cm. Find volume.

$$V=\frac{1}{3}\pi r^{2}h+\frac{2}{3}\pi r^3$$

$$=\frac{1}{3}\times \frac{22}{7}\times 49\times 24+\frac{2}{3}\times \frac{22}{7}\times 343$$

$$=1232+286.7=1518.7{\text{cm}}^3$$

Q5. A capsule is cylindrical with hemispherical ends. Length = 10 cm, radius = 1.5 cm. Find volume. Cylinder height = 10 – 2×1.5 = 7 cm.

$$V=\pi r^{2}h+\frac{4}{3}\pi r^3$$

$$=\frac{22}{7}\times (1.5{)}^2\times 7+\frac{4}{3}\times \frac{22}{7}\times (1.5{)}^3$$

$$=49.5+14.1=63.6{\text{cm}}^3$$

Q6. A tent is cylindrical up to height 2.8 m and conical above with height 2 m. Radius = 2 m. Find volume.

$$V=\pi r^2{h}_{cyl}+\frac{1}{3}\pi r^2{h}_{cone}$$

$$=\frac{22}{7}\times 4\times 2.8+\frac{1}{3}\times \frac{22}{7}\times 4\times 2$$

$$=35.2+18.7=53.9{\text{m}}^3$$

Q7. A capsule is cylindrical with hemispherical ends. Length = 7 cm, radius = 1 cm. Find volume. Cylinder height = 7 – 2×1 = 5 cm.

$$V=\pi r^{2}h+\frac{4}{3}\pi r^3$$

$$=\frac{22}{7}\times 1\times 5+\frac{4}{3}\times \frac{22}{7}\times 1$$

$$=15.7+4.19=19.9{\text{cm}}^3$$

Q8. A toy is in form of a cone surmounted on a hemisphere. Radius = 3.5 cm, height of cone = 7 cm. Find volume.

$$V=\frac{1}{3}\pi r^{2}h+\frac{2}{3}\pi r^3$$

$$=\frac{1}{3}\times \frac{22}{7}\times (3.5{)}^2\times 7+\frac{2}{3}\times \frac{22}{7}\times (3.5{)}^3$$

$$=89.6+89.6=179.2{\text{cm}}^3$$

Q9. A capsule is cylindrical with hemispherical ends. Length = 14 cm, radius = 3.5 cm. Find volume. Cylinder height = 14 – 2×3.5 = 7 cm.

$$V=\pi r^{2}h+\frac{4}{3}\pi r^3$$

$$=\frac{22}{7}\times (3.5{)}^2\times 7+\frac{4}{3}\times \frac{22}{7}\times (3.5{)}^3$$

$$=269.5+179.6=449.1{\text{cm}}^3$$

Q10. A tent is cylindrical up to height 3 m and conical above with height 2 m. Radius = 2.1 m. Find volume.

$$V=\pi r^2{h}_{cyl}+\frac{1}{3}\pi r^2{h}_{cone}$$

$$=\frac{22}{7}\times (2.1{)}^2\times 3+\frac{1}{3}\times \frac{22}{7}\times (2.1{)}^2\times 2$$

$$=41.6+26.4=68{\text{m}}^3$$

FAQs (10)

FAQ1. What is combination of solids? Object formed by joining two or more basic solids.

FAQ2. How to find volume of combination? Add volumes of individual solids.

FAQ3. How to find TSA of combination? Add exposed surface areas only.

FAQ4. What is CSA of cone? $\pi rl$.

FAQ5. What is TSA of hemisphere? $3\pi r^2$.

FAQ6. Why use Pythagoras theorem?

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