Chapter Notes: Volume

What is Volume?

• Volume is the amount of space inside an object, for example inside a box or a ball.
• We measure volume in cubic units because it involves three dimensions: length, width (or breadth) and height.
• Flat objects, such as book pages or a blackboard, do not have volume.
• For flat objects we measure area, not volume.

Look at the following solid shapes:

Solids have three dimensions - length, breadth and height.

For example:

• A glass has space to hold water.
• A box has space to hold objects.
• A balloon has space to hold air.

The space inside a solid is known as the volume of that solid.

The volume of a solid is the amount of space enclosed by it or the amount of space it takes up.

The other units used for measuring volume are cubic millimetre (mm³) and cubic metre (m³). The unit chosen depends on the size of the solid being measured.

Measuring Volume

• A 1 cm cube is called a unit cube and has volume 1 cubic centimetre written as 1 cm³.
• To measure the volume of a solid, we count how many unit cubes can fill it.
• When solids are made from unit cubes arranged in layers, the volume can be found by counting cubes per layer and multiplying by the number of layers.

Example 1: Find the volume of these solids by counting the number of cubes in each solid. The volume of each small cube is 1 cm³.

Sol: (a) There are 5 cubes in the given solid, so, volume = 5 cm³.
(b) 1st row = 8 cubes; 2nd row = 8 cubes;  3rd row = 8 cubes; 4th row = 8 cubes; 5th row = 8 cubes
∴ Volume = 8 cm³ + 8 cm³ + 8 cm³ + 8 cm³ + 8 cm³ = 40 cm³.
We can also say that there are 5 layers of cubes and each layer consists of 8 cubes. So, Volume = Number of layers × Number of cubes in each layer
V = (5 × 8) cm3 = 40 cm³.
(c) Cubes in horizontal row = 4 cubes Cubes in vertical row = 2 cubes
Volume = 4 cm³ x 2 cm³ = 8 cm³.

Volume of a Cuboid

• A cuboid is a solid made by stacking rectangular faces; it has length, width (breadth) and height.
• The volume of a cuboid is the amount of space inside it and is measured in cubic units (for example, cm³ or m³).
• A cuboid has 6 faces, 12 edges and 8 vertices (corners).
• If length, width or height changes, the volume changes accordingly.

If a cuboid is made of layers of equal rectangular sheets, the volume equals the area of the base rectangle multiplied by the height (number of layers times the thickness of each layer).

Formula for Volume of a Cuboid

• The area of the base is length × width.
• To get the volume, multiply the area of the base by the height.
• So, Volume = Length × Width × Height, which can be written as V = l × b × h.

Volume of Cuboid 

Examples

Example 1: See how a cuboid is built from 1-centimetre cubes and how we can find its volume.

Sol:

Example 2: The given cuboids are built from 1-centimetre cubes. Find the volume of each solid.

Sol: Use counting or use rows × columns × layers as shown in the table below.

Example 3: Find the volume of a cuboid whose length, breadth and height are 15 cm, 11 cm and 6 cm, respectively.

Sol: Length of the cuboid = 15 cm
Breadth of the cuboid = 11 cm
Height of the cuboid = 6 cm
Volume of the cuboid = length × breadth × height

Volume = 15× 11 × 6=990cm³

Volume of a Cube

• The volume of a cube is the total space inside it.
• A cube has six square faces, and all edges (sides) are equal in length.
• Another name for a cube is a regular hexahedron.
• Volume is measured in cubic units.

Cube

Formula for Volume of a Cube

• To find the volume of a cube, multiply the length of one side by itself three times; this is called "cubing" the side.
• So, if the side length is s,  Volume = side length × side length × side length, or simply V = s³.

Solved Examples for Volume of Cube

Example 1: Find the volume of a cube whose one edge measures 2 cm.

Sol: Length of each side of the cube = 2 cm.
Volume of the cube = side × side × side
= (2 × 2 × 2) cm³ = 8 cm³

Example 2: A box is 20 cm by 18 cm and 40 mm thick. How many cubic centimetres of space will the books occupy?

Sol: Length = 20 cm
Breadth = 18 cm
Height = 40 mm = (40 ÷ 10) cm = 4 cm
∴ Volume of the box = (20 × 18 × 4) cm³ = 1440 cm³
Hence, the books will occupy 1440 cm³ of space.

Capacity

Capacity of a container is the amount of a solid, liquid or gas it can hold.

For example:

• The capacity of a swimming pool is the amount of water needed to fill it (often measured in m³).
• The capacity of a gas cylinder is the amount of gas it can hold.
• The capacity of a vehicle's fuel tank is the amount of fuel needed to fill it.

• The common unit of capacity for liquids is the litre (L). Smaller amounts use millilitre (mL).

EduRev Tips:

• 1 litre = 1000 mL = 1000 cm³, 1 mL = 1 cm³
• 1 cm³ = 1 mL,
• 1 m³ = 100 cm × 100 cm × 100 cm
  = 1000000 cm³
  = 1000000 mL = 1000 L

Examples

Example 1: A rectangular tank measures 2.5 m by 3 m by 4 m and is full of water. What is its capacity in litres?

Sol: The tank measures 2.5 m by 3 m by 4 m, i.e., 250 cm by 300 cm by 400 cm.
Volume of the tank = (250 × 300 × 400) cm³
Since 1 litre = 1000 cm³,
So, capacity in litres = 250 × 300 × 400/1000 L = 30000 L.

Example 2: A machine for making ice freezes 5.76 litres of water into ice bricks measuring 3 cm by 2 cm by 1 cm. How many ice bricks will be made?

Sol: Volume of 1 ice brick = (3 × 2 × 1) cm³ = 6 cm³
Volume of water = 5.76 litres = 5.76 × 1000 cm3 = 5760 cm³
∴ Number of ice bricks made = 5760 ÷ 6 = 5760/6 = 960.

Example 3: There are 1.75 litres of water in the rectangular container shown. How much more water is needed to fill the container completely?

Sol: Capacity of the rectangular container = 15 cm × 10 cm × 20 cm
= 3000 cm³ = 3 L (Q 1000 cm³ = 1 L)
Volume of water in the container = 1.75 L
∴ Volume of water needed to fill the container
= 3 L - 1.75 L = 1.25 L
= (1.25 × 1000) mL = 1250 mL.