A closed cuboidal box has length $12$ cm, breadth $9$ cm and height $5$ cm. Find its total surface area.

**Given:** $l = 12$ cm, $b = 9$ cm, $h = 5$ cm. **Formula:** Total surface area of a cuboid $= 2(lb + bh + hl)$. **Substitute:** $$= 2(12\times 9 + 9\times 5 + 5\times 12)$$ $$= 2(108 + 45 + 60)$$ $$= 2 \times 213 = 426 \text{ cm}^2$$ **Total surface area $= 426$ cm².**

The side of a cube is $7$ cm. Find (i) its lateral surface area and (ii) its total surface area.

**Given:** edge $a = 7$ cm. **(i) Lateral surface area** (four side faces) $= 4a^2$: $$= 4 \times 7^2 = 4 \times 49 = 196 \text{ cm}^2$$ **(ii) Total surface area** (all six faces) $= 6a^2$: $$= 6 \times 49 = 294 \text{ cm}^2$$ **LSA $= 196$ cm², TSA $= 294$ cm².**

A cylindrical pillar has diameter $50$ cm and height $3.5$ m. Find the cost of painting its curved surface at ₹$12$ per m². (Use $\pi = \tfrac{22}{7}$.)

**Given:** diameter $= 50$ cm $\Rightarrow r = 25$ cm $= 0.25$ m; height $h = 3.5$ m. **Curved surface area** $= 2\pi r h$: $$= 2 \times \frac{22}{7} \times 0.25 \times 3.5$$ $$= 2 \times \frac{22}{7} \times 0.875 = \frac{38.5}{7} \times ... $$ Let us compute step by step: $$= \frac{44}{7} \times 0.25 \times 3.5 = \frac{44 \times 0.875}{7} = \frac{38.5}{7} = 5.5 \text{ m}^2$$ **Cost** $= 5.5 \times 12 = ₹66$. **Cost of painting $= ₹66$.**

A cylindrical vessel open at the top has base radius $7$ cm and height $10$ cm. Find the total area of metal sheet needed to make it. (Use $\pi = \tfrac{22}{7}$.)

**Given:** $r = 7$ cm, $h = 10$ cm. The vessel is open at the top, so it has curved surface + one base. **Metal sheet** $= 2\pi r h + \pi r^2$. Curved surface $= 2 \times \frac{22}{7} \times 7 \times 10 = 2 \times 22 \times 10 = 440$ cm². Base $= \frac{22}{7} \times 7^2 = 22 \times 7 = 154$ cm². **Total** $= 440 + 154 = 594$ cm². **Metal sheet required $= 594$ cm².**

A cuboidal room is $8$ m long, $6$ m wide and $4$ m high. Find the area of the four walls that must be painted.

**Given:** $l = 8$ m, $b = 6$ m, $h = 4$ m. The four walls form the lateral surface area $= 2(l + b)h$. **Substitute:** $$= 2(8 + 6)\times 4 = 2 \times 14 \times 4 = 112 \text{ m}^2$$ **Area of four walls $= 112$ m².**

Find the volume of a cone whose base radius is $6$ cm and height is $14$ cm. (Use $\pi = \tfrac{22}{7}$.)

**Given:** $r = 6$ cm, $h = 14$ cm. **Formula:** Volume of a cone $= \tfrac{1}{3}\pi r^2 h$. **Substitute:** $$= \frac{1}{3} \times \frac{22}{7} \times 6^2 \times 14$$ $$= \frac{1}{3} \times \frac{22}{7} \times 36 \times 14$$ $$= \frac{1}{3} \times 22 \times 36 \times 2 = \frac{1584}{...}$$ Compute: $\frac{22}{7}\times 14 = 44$, so $= \frac{1}{3}\times 44 \times 36 = \frac{1584}{3} = 528$ cm³. **Volume $= 528$ cm³.**

Find the volume of a sphere of radius $3.5$ cm. (Use $\pi = \tfrac{22}{7}$.)

**Given:** $r = 3.5$ cm $= \tfrac{7}{2}$ cm. **Formula:** Volume of sphere $= \tfrac{4}{3}\pi r^3$. **Substitute:** $$= \frac{4}{3} \times \frac{22}{7} \times \left(\frac{7}{2}\right)^3$$ $$= \frac{4}{3} \times \frac{22}{7} \times \frac{343}{8}$$ $$= \frac{4}{3} \times \frac{22 \times 49}{8} = \frac{4}{3} \times \frac{1078}{8} = \frac{4312}{24} \approx 179.67 \text{ cm}^3$$ **Volume $\approx 179.67$ cm³.**

A hemispherical bowl has inner radius $10.5$ cm. Find the volume of water it can hold. (Use $\pi = \tfrac{22}{7}$.)

**Given:** $r = 10.5$ cm $= \tfrac{21}{2}$ cm. **Formula:** Volume of hemisphere $= \tfrac{2}{3}\pi r^3$. **Substitute:** $$= \frac{2}{3} \times \frac{22}{7} \times \left(\frac{21}{2}\right)^3$$ $$= \frac{2}{3} \times \frac{22}{7} \times \frac{9261}{8}$$ $$= \frac{2}{3} \times \frac{22 \times 1323}{8} = \frac{2}{3} \times \frac{29106}{8}$$ $$= \frac{2 \times 29106}{24} = \frac{58212}{24} = 2425.5 \text{ cm}^3$$ **Volume of water $= 2425.5$ cm³.**

A conical tent has base radius $7$ m and height $24$ m. Find its capacity in litres. (Use $\pi = \tfrac{22}{7}$; $1$ m³ $= 1000$ L.)

**Given:** $r = 7$ m, $h = 24$ m. **Volume** $= \tfrac{1}{3}\pi r^2 h$: $$= \frac{1}{3} \times \frac{22}{7} \times 7^2 \times 24$$ $$= \frac{1}{3} \times \frac{22}{7} \times 49 \times 24$$ $$= \frac{1}{3} \times 22 \times 7 \times 24 = \frac{1}{3} \times 3696 = 1232 \text{ m}^3$$ **Convert:** $1232 \times 1000 = 1\,232\,000$ litres. **Capacity $= 1{,}232{,}000$ litres.**

A solid metallic sphere of radius $6$ cm is melted and recast into small cones each of radius $2$ cm and height $3$ cm. How many cones are formed?

**Key idea:** Volume is conserved on melting and recasting. **Volume of sphere** $= \tfrac{4}{3}\pi r^3 = \tfrac{4}{3}\pi (6)^3 = \tfrac{4}{3}\pi \times 216 = 288\pi$ cm³. **Volume of one cone** $= \tfrac{1}{3}\pi r^2 h = \tfrac{1}{3}\pi (2)^2 (3) = \tfrac{1}{3}\pi \times 12 = 4\pi$ cm³. **Number of cones** $= \dfrac{\text{Volume of sphere}}{\text{Volume of one cone}} = \dfrac{288\pi}{4\pi} = 72$. **72 cones are formed.**

A cylindrical tank of radius $1.4$ m and height $3$ m is full of water. Find how many litres it holds. (Use $\pi = \tfrac{22}{7}$.)

**Given:** $r = 1.4$ m, $h = 3$ m. **Volume** $= \pi r^2 h$: $$= \frac{22}{7} \times (1.4)^2 \times 3$$ $$= \frac{22}{7} \times 1.96 \times 3$$ $$= \frac{22}{7} \times 5.88 = 22 \times 0.84 = 18.48 \text{ m}^3$$ **Convert:** $18.48 \times 1000 = 18\,480$ litres. **The tank holds $18{,}480$ litres.**

CBSE • Class 9 • Mathematics • Chapter 11

Surface Areas and Volumes — NCERT Solutions, AI Tutor & Practice

Surface area and volume of standard solids — cuboid, cube, right circular cylinder, right circular cone, sphere and hemisphere.

Aligned to the latest NCERT 2024-25 edition • 2 exercises covered • Free plan, no credit card

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What you will learn

Compute curved surface area, total surface area and volume of cuboid, cube and cylinder
Compute curved surface area, total surface area and volume of cone, sphere and hemisphere
Apply formulas to real-world problems on tanks, balls, traffic cones and similar

Key concepts in this chapter

CuboidCubeCylinderConeSphereHemisphere

NCERT Exercise-wise Solutions

2 exercises • 20 questions covered • Open any exercise for theme, focus topics and Socratic AI walkthroughs

Exercise 12.1 • 9 Qs

Surface areas of cuboid, cube, cylinder and combinations.

CSA vs TSACylinder wrapComposite solids

Exercise 12.2 • 11 Qs

Volumes of solids — cones, spheres, hemispheres and capacity conversions.

Volume formulasLitres conversionMixed solids

Frequently asked NCERT questions in this chapter

Find the curved surface area of a cone of radius 7 cm and slant height 25 cm.
Find the volume of a sphere of radius 10.5 cm.
A cuboidal water tank is 1.5 m by 1 m by 0.8 m. How many litres can it hold?

Step-by-step NCERT solutions

11 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 12.1 Q1 • 2 marks

12

Hint (Socratic — try this first)▾

Which formula adds up all six rectangular faces of a cuboid?

Step-by-step solution▾

l = 12

Common mistake:

Forgetting to multiply the bracket by 2 (which would give the area of only three faces).

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Exercise 12.1 Q2 • 2 marks

7

Hint (Socratic — try this first)▾

How many faces make up the lateral surface, and how many make the total surface?

Step-by-step solution▾

a = 7

Common mistake:

4a^2

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Exercise 12.1 Q3 • 3 marks

50

Hint (Socratic — try this first)▾

What is the radius in metres, and which surface area do you paint on a pillar?

Step-by-step solution▾

= 50

Common mistake:

Using diameter instead of radius, or forgetting to convert centimetres to metres before computing.

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Exercise 12.1 Q4 • 3 marks

7

Hint (Socratic — try this first)▾

An open cylinder has a curved surface plus how many circular ends?

Step-by-step solution▾

r = 7

Common mistake:

Including both circular ends (as in a closed cylinder) instead of only one, since the top is open.

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Exercise 12.1 Q5 • 2 marks

8

Hint (Socratic — try this first)▾

Which surface area of a cuboid covers only the walls and not the floor or ceiling?

Step-by-step solution▾

l = 8

Common mistake:

Using the total surface area formula, thereby wrongly including the floor and ceiling.

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Exercise 12.2 Q1 • 2 marks

6

Hint (Socratic — try this first)▾

How does the volume of a cone compare with that of a cylinder of the same base and height?

Step-by-step solution▾

r = 6

Common mistake:

\tfrac{1}{3}

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Exercise 12.2 Q2 • 3 marks

3.5

Hint (Socratic — try this first)▾

Which power of the radius appears in the sphere's volume formula?

Step-by-step solution▾

r = 3.5

Common mistake:

r^2

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Exercise 12.2 Q3 • 3 marks

10.5

Hint (Socratic — try this first)▾

A hemisphere is what fraction of a full sphere?

Step-by-step solution▾

r = 10.5

Common mistake:

\tfrac{4}{3}\pi r^3

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Exercise 12.2 Q4 • 3 marks

7

Hint (Socratic — try this first)▾

Once you find the volume in cubic metres, how do you convert to litres?

Step-by-step solution▾

r = 7

Common mistake:

\tfrac{1}{3}

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Exercise 12.2 Q5 • 3 marks

6

Hint (Socratic — try this first)▾

When one solid is recast into another, which quantity stays the same?

Step-by-step solution▾

= \tfrac{4}{3}\pi r^3 = \tfrac{4}{3}\pi (6)^3 = \tfrac{4}{3}\pi \times 216 = 288\pi

Common mistake:

\pi

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Exercise 12.2 Q6 • 3 marks

1.4

Hint (Socratic — try this first)▾

What is the volume formula for a cylinder, and how do cubic metres relate to litres?

Step-by-step solution▾

r = 1.4

Common mistake:

Squaring the diameter instead of the radius, or forgetting to convert m³ to litres.

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FAQs about this chapter

What is the difference between curved surface area and total surface area of a cylinder?+

Curved surface area is just the lateral curved part (2πrh). Total surface area additionally includes the two flat circular ends, so it equals 2πrh + 2πr².

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Related chapters

Chapter 10

Heron's Formula

Chapter 12

Statistics

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