ICSE Class 8 Maths Chapter 33 Volume and Surface Area Cuboid Cube

Chapter 33: Volume And Surface Area

Cuboid and Cube

33.1 Introduction

Also,

1 m³ = 100 × 100 × 100 cm³ = 1000000 cm³ and 1 cm³ = \(\frac{1}{100 \times 100 \times 100}\) m³

1 cm³ = 10 × 10 × 10 mm³ = 1000 mm³ and 1 mm³ = \(\frac{1}{1000}\) cm³

In general, the volume of a liquid or a gas is measured in litres, such that 1 m³ = 1000 litre and 1 litre = 1000 cm³ (c.c. or millilitre)

Teacher's Note

Understanding volume and surface area is essential in everyday life - from calculating how much water a tank can hold to determining the amount of paint needed to cover a wall.

33.2 Cuboid (A Rectangular Solid)

A cuboid is a solid bounded by six rectangular faces.

1. Volume of a cuboid

= its length × breadth × height

= l × b × h

2. Total surface area of a cuboid

= Area of six rectangular faces

Since, Area of ABCD + Area of EFGH = 2(l × b) [Opposite faces are equal]

Area of BCGF + Area of ADHE = 2(b × h) [Opposite faces are equal]

and Area of ABFE + Area of DCGH = 2(h × l) [Opposite faces are equal]

Therefore, Total surface area of cuboid = 2(l × b + b × h + h × l)

Teacher's Note

Cuboids are found everywhere - from storage boxes to refrigerators - making these formulas practical for calculating storage capacity and material costs.

33.3 Cube

A cube is a rectangular solid whose each face is a square. In other words, a cube is a cuboid whose length = breadth = height = a (say)

1. Volume of a cube

Since volume of a cuboid = l × b × h

Therefore, Volume of a cube = a × a × a = a³ = (its edge)³

2. Total surface area of a cube

= 2 (a × a + a × a + a × a) = 6a² = 6 (edge)²

Teacher's Note

Cubes are special cases of cuboids with equal dimensions, making calculations simpler and appearing frequently in dice, building blocks, and storage containers.

Example 1

The length, breadth and height of a cuboid are in the ratio 6 : 5 : 4. If its volume is 15,000 cm³; find: (i) its dimensions (ii) its surface area.

Solution

Dimension means: Its length, breadth and height.

(i) Given: Length : breadth : height = 6 : 5 : 4

If length = 6x cm, breadth = 5x cm and height = 4x cm

Therefore, Length × breadth × height = volume

6x × 5x × 4x = 15,000

x³ = \(\frac{15,000}{6 \times 5 \times 4}\) = 125 = 5 × 5 × 5 = 5³

x = 5

i.e. length = 6x cm = 6 × 5 cm = 30 cm

breadth = 5x cm = 5 × 5 cm = 25 cm

and, height = 4x cm = 4 × 5 cm = 20 cm (Ans.)

(ii) Surface area of the cuboid = 2(l × b + b × h + h × l)

= 2(30 × 25 + 25 × 20 + 20 × 30) cm²

= 2(750 + 500 + 600) cm² = 3700 cm² (Ans.)

Example 2

The total surface area of a cube is 294 cm², find its volume.

Solution

Since total surface area of cube = 6 × (side)²

6 × (side)² = 294

side = 7 cm

Therefore, volume = (side)³ = (7 cm)³ = 343 cm³ (Ans.)

Example 3

A rectangular solid of metal has dimensions 50 cm, 64 cm and 72 cm. It is melted and recast into identical cubes each with edge 4 cm, find the number of cubes formed.

Solution

Therefore, Volume of rectangular solid melted = its length × breadth × height = 50 × 64 × 72 cm³

And, volume of each cube formed = (its edge)³ = (4)³ cm³ = 4 × 4 × 4 cm³

Therefore, Number of cubes formed = \(\frac{\text{Volume of solid melted}}{\text{Volume of each cube}}\)

= \(\frac{50 \times 64 \times 72}{4 \times 4 \times 4}\) = 3600 (Ans.)

Example 4

Three cubes, each of edge 8 cm, are joined as shown alongside. Find the total surface area and the volume of the cuboid.

Solution

Since, length (l) of the resulting cuboid = 3 × 8 cm = 24 cm, its breadth (b) = 8 cm and its height (h) = 8 cm

Total surface area = 2(l × b + b × h + h × l)

= 2(24 × 8 + 8 × 8 + 8 × 24) cm² = 896 cm² (Ans.)

Volume = l × b × h

= 24 × 8 × 8 cm³ = 1536 cm³ (Ans.)

Teacher's Note

When multiple identical cubes are arranged together, they form a cuboid with dimensions that depend on how many cubes are placed along each direction.

Test Yourself

1. 1 m = ............... cm, 1 m² = ............... × ............... cm² = .................. cm² and 1 m³ = ............... × ............... × ............... cm³ = ........................... cm³.

2. 1 m³ = ................... litre and 1 litre = .................. cm³.

3. A cube is always a .................. , but a cuboid is not necessarily a ....................

4. The volume of a cube with side a cm is numerically equal to its surface area; then .................. = .................. and a = ....................

5. Each edge of a cube is 8 cm; area of each face of the cube = .............. × .............. cm² = .............. cm² and total surface area of the cube is .................. cm².

6. Each edge of a cube is doubled, then its total surface area becomes .................. times and its volume becomes .................. times.

7. A solid cuboid (36 cm × 3 cm × x cm) has the same volume as a solid cube of edge 6 cm; then .......................... = .......................... and x = .................. = ....................

8. A cubical container, with each edge 10 cm, is full of water. This water is transferred to an empty rectangular container with length 20 cm and breadth 5 cm. If the height of water in the rectangular container is x cm, then 10 × 10 × 10 = .................. and x = ..........................................

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