Volume and Capacity - Mathematics Form 1 Notes

• Introduction
  • Conversion of Units of Volume
• Volume of Cubes, Cuboids and Cylinders
  • Cube
  • Cuboid
  • Cylinder
• Capacity
  • Conversion of Units of Capacity
  • Relationship between Volume and Capacity
• Past KCSE Questions on the Topic

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

1. 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.
2. 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:
   1. The total number of fans airlifted to South Africa. (2mks)
   2. The total cost of fuel used if one litre costs 0.3 dollars. (4mks)
   3. The total collection in dollars made by each plane. (2mks)
   4. The net profit made by each plane. (2mks)
3. 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
4. Three litres of water (density 1 g/cm³) is added to twelve litres of alcohol (density 0.8g/cm³). What is the density of the mixture?
5. A rectangular tank whose internal dimensions are 2.2m by 1.4m by 1.7m is three fifth full of milk.
   1. Calculate the volume of milk in litres
   2. 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:
      1. The volume in cm3 of each packet to the nearest whole number
      2. The number of full packets of milk
      3. The amount of money realized from the sale of milk
6. 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/m³ correct to 1 decimal place.