CBSE Class 10 Mathematics NCERT Solutions Chapter 13: Surface Areas and Volumes

Reviewed by NCERT Help Editorial Team · Updated on 29 Jul 2026

NCERT Solutions PDF Class 10 PDF

This chapter provides comprehensive NCERT Solutions for Class 10 Mathematics, focusing on Chapter 13: Surface Areas and Volumes. Students will learn to calculate the surface areas of combined solid figures. The solutions cover problems involving the joining of cubes and cylinders with hemispheres, detailing step-by-step calculations for surface areas. Key concepts include understanding the dimensions of composite shapes and applying formulas for surface areas of cuboids, cylinders, and hemispheres. These solutions are designed to help students grasp the concepts thoroughly and prepare effectively for their board examinations by offering clear explanations and accurate results.

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Quick info

Board	CBSE
Class	Class 10
Subject	Mathematics
Session	2026
Language	English
Type	NCERT Solutions
Chapter	Chapter 13: Surface Areas and Volumes

Chapter summary

Chapter 13 of the NCERT Class 10 Mathematics textbook deals with Surface Areas and Volumes of combined solids. This section provides detailed solutions to problems involving composite shapes formed by joining basic solids like cubes, cylinders, and hemispheres. Students will practice calculating the surface area of these combined structures, understanding how the dimensions change and which formulas to apply. The exercises focus on practical applications and reinforce the understanding of geometric formulas.

Learning outcomes

Understand the concept of surface area for combined solid figures.
Calculate the edge length of a cube from its volume.
Determine the dimensions of a cuboid formed by joining cubes.
Calculate the surface area of a cuboid.
Identify the radius and height of composite shapes like a cylinder mounted on a hemisphere.
Calculate the inner surface area of a vessel composed of a cylinder and a hemisphere.

Topics covered

Paper topics

Surface area of combined solids
Cuboids
Cubes
Hemispheres
Cylinders
Volume of cubes
Surface area of cuboids
Surface area of cylinders
Surface area of hemispheres
Composite shapes

Important topics

Surface area of combined solids
Calculating dimensions of composite shapes
Surface area of cuboids
Surface area of cylinder mounted on hemisphere

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Questions and Solutions

Question 1

Two cubes, each with a volume of 64 cm³, are joined end to end. Find the surface area of the resulting cuboid.

Solution:

Given the volume of each cube is 64 cm³.

Let the edge length of one cube be 'a' cm.

The volume of a cube is given by the formula $V = a^3$ .

So, $a^3 = 64 \text{ cm}^3$ .

Taking the cube root of both sides, we get $a = \sqrt[3]{64} = 4 \text{ cm}$ .

When two such cubes are joined end to end, they form a cuboid. The dimensions of this cuboid will be:

Length (l) = a + a = 4 cm + 4 cm = 8 cm
Breadth (b) = a = 4 cm
Height (h) = a = 4 cm

The surface area of a cuboid is given by the formula $A = 2(lb + bh + lh)$ .

Substituting the dimensions:

$A = 2((8 \times 4) + (4 \times 4) + (8 \times 4))$

$A = 2(32 + 16 + 32)$

$A = 2(80)$

$A = 160 \text{ cm}^2$

Therefore, the surface area of the resulting cuboid is 160 cm².

Question 2

A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. Use $\pi = \frac{22}{7}$ .

Solution:

The vessel consists of a hollow cylinder mounted on a hollow hemisphere.

Given the diameter of the hemisphere is 14 cm.

The radius (r) of the hemisphere is half of the diameter: $r = \frac{14}{2} = 7 \text{ cm}$ .

Since the cylinder is mounted on the hemisphere, the radius of the cylinder is also the same as the radius of the hemisphere, so $r = 7 \text{ cm}$ .

The total height of the vessel is 13 cm.

The height of the hemispherical part is equal to its radius, which is 7 cm.

The height of the cylindrical part (h) is the total height minus the height of the hemispherical part:

$h = \text{Total height} - \text{Radius of hemisphere}$

$h = 13 \text{ cm} - 7 \text{ cm} = 6 \text{ cm}$ .

The inner surface area of the vessel is the sum of the curved surface area (CSA) of the cylindrical part and the curved surface area of the hemispherical part.

CSA of cylinder = $2\pi rh$

CSA of hemisphere = $2\pi r^2$

Inner surface area = $2\pi rh + 2\pi r^2$

Substituting the values $r = 7 \text{ cm}$ , $h = 6 \text{ cm}$ , and $\pi = \frac{22}{7}$ :

Inner surface area = $\left(2 \times \frac{22}{7} \times 7 \times 6\right) + \left(2 \times \frac{22}{7} \times 7 \times 7\right)$

Inner surface area = $(2 \times 22 \times 6) + (2 \times 22 \times 7)$

Inner surface area = $264 + 308$

Inner surface area = $572 \text{ cm}^2$

Therefore, the inner surface area of the vessel is 572 cm².

Common mistakes

Incorrectly calculating the dimensions of the resulting cuboid after joining cubes.
Forgetting to add the surface areas of individual components when calculating the total surface area of a composite shape.
Using the wrong formula for the surface area of a cylinder or hemisphere.
Errors in calculating the height of the cylindrical part when it's part of a composite vessel.

Revision tips

Visualize how cubes are joined to form a cuboid and how this affects its dimensions.
Clearly identify the individual shapes that make up a composite solid and their respective dimensions.
Practice applying the correct surface area formulas for each component shape.
Pay close attention to the given values, especially the diameter and total height, to correctly derive individual dimensions.

Practice MCQs

Q1. When two cubes of edge length 4 cm are joined end to end, what are the dimensions of the resulting cuboid?

Q2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. If the radius of the hemisphere is 7 cm and the height of the cylindrical part is 6 cm, what is the curved surface area of the cylindrical part?

Q3. What is the surface area of a cuboid formed by joining two cubes, each with a volume of 64 c $m^3$ ?

Q4. In a vessel formed by a cylinder on a hemisphere, the radius is 7 cm and the total height is 13 cm. What is the height of the hemispherical part?

Frequently asked questions

What is Chapter 13 of CBSE Class 10 Maths about?

Chapter 13 of CBSE Class 10 Maths covers Surface Areas and Volumes, focusing on calculating the surface areas of solid figures formed by combining basic shapes like cubes, cylinders, and hemispheres.

How are the solutions for combined solids calculated?

The solutions involve identifying the individual shapes, determining their dimensions (often by subtracting or equating parts), and then summing the relevant surface areas, ensuring no overlapping areas are counted twice.

What is the surface area of a cuboid formed by joining two cubes?

If two cubes of edge 'a' are joined end-to-end, they form a cuboid with dimensions a x a x 2a. The surface area is calculated using the cuboid formula: 2(a*a + a*2a + a*2a).

How do I find the surface area of a vessel made of a cylinder and a hemisphere?

The inner surface area is the sum of the curved surface area of the cylinder (2πrh) and the curved surface area of the hemisphere (2πr^2), where 'r' is the common radius and 'h' is the height of the cylinder.

Are the questions in these NCERT solutions similar to exam questions?

Yes, these NCERT solutions cover fundamental concepts and problem types frequently tested in CBSE Class 10 Mathematics exams, particularly those involving composite shapes.

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