## Introduction

- Volume is the amount of space occupied by a solid object. The unit of volume is cubic units.
- A cube of edge 1 cm has a volume of 1 cm x 1 cm x 1 cm = 1 cm³.

### Conversion of Units of Volume

- A cube of side 1 m has a volume of 1 m³

But 1 m = 100 cm

1 m x 1 m x 1 m = 1 00 cm x 1 00 cm x 1 00 cm

Thus, 1 m = (0.01 x 0.01 x 0.01) m³

=0.000001 m³

=1 x 1 0¯⁶m³ - A cube side 1 cm has a volume of 1 cm³.

But 1 cm=1 0mm

1 cm x 1 cm x 1 cm = 1 0 mm x 1 0 mm x 1 0 mm

Thus, 1 cm³= 1 000mm³

## Volume of Cubes, Cuboids and Cylinders

### Cube

- A cube is a solid having six plane square faces in which the angle between two adjacent faces is a right-angle.

Volume of a cube= area of base x height

=l²x l

=l³

### Cuboid

- A cuboid is a solid with six faces which are not necessarily square.

Volume of a cuboid = length x width x height

= a sq. units x h

= ah cubic units.

### Cylinder

- This is a solid with a circular base.

Volume of a cylinder = area of base x height

=πr² x h

=πr²h cubic units

**Example**

Find the volume of a cuboid of length 5 cm, breadth 3 cm and height 4 cm.

**Solution**

Area of its base = 5x4 cm²

Volume =5 x 4 x 3 cm³

= 60 cm³

**Example**

Find the volume of a solid whose cross-section is a right- angled triangle of base 4 cm, height 5 cm and length 12 cm.

**Solution**

Area of cross-section =^{1}/_{2} x4 x 5

=10 cm²

Therefore volume =1 0 x 1 2

=120 cm³

**Example**

Find the volume of a cylinder with radius 1 .4 m and height 13 m.

**Solution**

Area of cross-section= 22/7 x 1.4 x 1.4

=6.16 m²

Volume = 6.1 6 x 1 3

=80.08 m³

In general, volume v of a cylinder of radius r and length l given by v=πr²l

## Capacity

- Capacity is the ability of a container to hold fluids. The SI unit of capacity is litre (l)

### Conversion of Units of Capacity

1 centiliter (cl)=10 millilitre (ml)

1 decilitre dl = 10 centilitre (cl)

1 litre (l) =10 decilitres (dl)

1 Decalitre (Dl) = 10 litres (l)

1 hectolitre (Hl) =10 decalitre( Dl)

1 kilolitre (kl) =10 hectolitres (Hl)

1 kilolitre (kl)= 1000 litres (l)

1 litre (l) =1000 millilitres (ml)

### Relationship Between Volume and Capacity

A cube of an edge 10 cm holds 1 litre of liquid.

1 litre =10 cm x 10cm x 10 cm

= 1000 cm³

1 m³ = 10⁶ cm³

1 m³ = 10³ litres.

## Past KCSE Questions on the Topic

- All the water is poured into a cylindrical container of circular radius 1 2cm. If the cylinder has height 45cm, calculate the surface area of the cylinder which is not in contact with water.
- The British government hired two planes to airlift football fans to South Africa for the World cup tournament. Each plane took 10½ hours to reach the destination. Boeng 747 has carrying capacity of 300 people and consumes fuel at 120 litres per minute. It makes 5 trips at full capacity. Boeng 740 has carrying capacity of 140 people and consumes fuel at 200 litres per minute. It makes 8 trips at full capacity. If the government sponsored the fans one way at the cost of 800 dollars per fan, calculate:
- The total number of fans airlifted to South Africa. (2mks)
- The total cost of fuel used if one litre costs 0.3 dollars. (4mks)
- The total collection in dollars made by each plane. (2mks)
- The net profit made by each plane. (2mks)

- A rectangular water tank measures 2.6m by 4.8m at the base and has water to a height of 3.2m. Find the volume of water in litres that is in the tank
- Three litres of water (density 1 g/cm
^{3}) is added to twelve litres of alcohol (density 0.8g/cm^{3}). What is the density of the mixture? - A rectangular tank whose internal dimensions are 2.2m by 1.4m by 1.7m is three fifth full of milk.
- Calculate the volume of milk in litres
- The milk is packed in small packets in the shape of a right pyramid with an equilateral base triangle of sides 1 0cm. The vertical height of each packet is 13.6cm. Full packets obtained are sold at shs.30 per packet. Calculate:
- The volume in cm3 of each packet to the nearest whole number
- The number of full packets of milk
- The amount of money realized from the sale of milk

- An 890kg culvert is made of a hollow cylindrical material with outer radius of 76cm and an inner radius of 64cm. It crosses a road of width 3m, determine the density of the material ssused in its construction in Kg/m
^{3 }correct to 1 decimal place.

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