NCERT Class 9 Ganita Manjari Part 2 Chapter 14 Math of Space Surface Area and Volume PDF Download

Chapter 14: Math of Space: Surface Area and Volume

In the previous grade we explored nets of some solids including cubes, cylinders, and cones. We saw that solids have a surface and occupy some space or volume. In this chapter, we will learn how to determine surface areas and volumes of cuboids, cylinders, cones, pyramids, and spheres, and how we can use these to solve some interesting problems.

14.1 Cuboids and Cubes

A cuboid has the shape shown in these pictures. A useful way of thinking about a cuboid is that it is a 'three-dimensional (3D) version' of a rectangle.

It has six faces, all rectangles. It is characterised by three quantities: length, width, and height - these are denoted by \( l \), \( w \), \( h \) respectively.

You can cut out the net of a cuboid given at the end of the textbook, make appropriate folds and join the edges to form the cuboid.

What is the total surface area of a cuboid? You may refer to the net of the cuboid given in Fig. 14.2.

Considering the faces individually, we see that the total surface area (TSA) of the cuboid is

\[ \text{TSA} = 2(wl + hl + hw). \]

[Figure 14.1A: Cuboid bricks, See in your textbook]

[Figure 14.1B: Rubik's Cube, See in your textbook]

[Figure 14.1C: Cuboid, See in your textbook]

[Figure 14.2: Net of a cuboid, See in your textbook]

Volume measures the amount of three-dimensional space an object or a container occupies. It is measured in cubic units, e.g., cubic meters (\( m^3 \)) or cubic centimeters (\( cm^3 \)). We set the volume of a unit cube (i.e., a cuboid measuring one unit in each direction) to be one cubic unit (e.g., 1 \( m^3 \) or 1 \( cm^3 \), as the case may be). The volume of a general cuboid is the number of unit cubes that fit into it. Accordingly, the formula for the volume \( V \) of a cuboid is length \( \times \) width \( \times \) height or

\[ V = lwh. \]

Compare the formula for the volume of a cuboid with the formula for the area of a rectangle (length \( \times \) width). Here is a way of understanding why the formula \( V = lwh \) is true: think of a cuboid as a large number of thin rectangular pieces (length \( l \), width \( w \)) stacked on top of each other, like a pack of playing cards. (Fig. 14.3).

[Figure 14.3: Pack of playing cards, See in your textbook]

Teacher's Note

The formula \( V = lwh \) is just the 2D area formula extended to 3D. Once you find the area of the base (length \( \times \) width), multiply by the height to stack those layers. This same pattern works for all prisms: volume = base area \( \times \) height.

A particular and important case of a cuboid is a cube, whose length, width, and height are the same. This length is the 'side' of the cube.

Let the side be \( a \). Then:

• Total surface area of the cube is \( 6a^2 \).
• Volume of the cube is \( a^3 \).

[Figure 14.4: Cube with side a, See in your textbook]

Think and Reflect

Try to work out for yourself why this model explains the formula for the volume of a cuboid, i.e.,

\[ \text{volume} = \text{area of base} \times \text{height}. \]

Write the formula for the surface area and the volume of a cube.

Observe how these two formulas are special cases of the formulas \( 2(wl + hl + hw) \) and \( lwh \), when \( w = h = l \).

Do you remember cube numbers? Cube numbers are the numbers in the sequence 1, 8, 27, 64, .... These numbers represent the number of unit cubes that can fit inside cubes of side lengths 1, 2, 3, 4, ... respectively - in other words, they represent the volumes of these cubes. We know that \( 1^3 = 1 \), \( 2^3 = 8 \), \( 3^3 = 27 \), \( 4^3 = 64 \), ...; and \( \sqrt[3]{1} = 1 \), \( \sqrt[3]{8} = 2 \), \( \sqrt[3]{27} = 3 \), \( \sqrt[3]{64} = 4 \), and so on.

Example 1

Example 1: Two solid objects are made from the same material: Cube A has side 6 cm and Cuboid B has dimensions 9 cm \( \times \) 6 cm \( \times \) 4 cm. Compare their volumes and surface areas and determine which object has a greater surface area. How is this useful in a real-life situation?

For Cube A, side (\( a \)) = 6 cm.

Hence its volume is \( a^3 = (6 \text{ cm})^3 = 216 \text{ cm}^3 \) and surface area (SA) is \( 6a^2 = 6 \times (6 \text{ cm})^2 = 216 \text{ cm}^2 \).

For Cuboid B, length (\( l \)) = 9 cm; width (\( w \)) = 6 cm; height (\( h \)) = 4 cm.

Hence its volume is \( l \times w \times h = 9 \times 6 \times 4 = 216 \text{ cm}^3 \)

and surface area (SA) is \( 2(lw + wh + lh) = 2[(9 \times 6) + (6 \times 4) + (4 \times 9)] = 228 \text{ cm}^2 \).

Thus, the two objects have the same volume but different surface areas. The cuboid has greater surface area than the cube.

Conclusion: For the same storage capacity: a cube requires less covering material; a cuboid exposes more area to its surroundings. This concept is important in packaging, heat transfer and engineering.

Teacher's Note

When two objects have the same volume, the one with the larger surface area will lose or gain heat faster. This is why cubes are more efficient for storage (less material needed) but cuboids spread heat faster. Always check both formulas separately - do not assume that if volumes are equal, surface areas must be equal too.

Exercise Set 14.1

1. The volume of a cube is 64 cm\( ^3 \). What is its total surface area?
2. How many small cubes with side 20 cm can be packed tight in a cubical box with side 2 m?
3. The dimensions of a godown are 40 m \( \times \) 25 m \( \times \) 10 m. If it is filled with cuboidal boxes, each of dimensions 2 m \( \times \) 1.25 m \( \times \) 1 m, then find the number of boxes.
4. Two cubes each of volume 125 cm\( ^3 \) are joined end to end. Find the surface area of the resulting cuboid.

Key Points

• A cuboid's total surface area is \( 2(wl + hl + hw) \) and its volume is \( V = lwh \), where \( l \), \( w \), and \( h \) are length, width, and height.
• A cube is a special cuboid where all sides are equal (\( a \)); its surface area is \( 6a^2 \) and volume is \( a^3 \).
• Two objects can have the same volume but different surface areas; this matters in real life for packaging, heat transfer, and how much material you need to cover something.

This is a preview of the first 3 pages. To get the complete chapter, click below.