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Volume and Capacity

Measurement · Volume and Capacity

Syllabus tag: KCSE | Mathematics | Form 1 | Topic 14 Volume and Capacity

Lesson objectives

By the end of this topic, you should be able to:

  • State the units of volume and convert from one to another.
  • Calculate the volume of cubes, cuboids and cylinders.
  • State the units of capacity and relate them to units of volume.
  • Solve problems involving volume and capacity in real situations.

Volume and Capacity

Volume is the space a solid occupies. Capacity is how much a container holds. They describe the same thing from two sides.

a) Units and converting

Converting volume units Converting volume units 1 m = 100 cm the length conversion 1 m³ = 100 × 100 × 100 cube ALL THREE sides 1 m³ = 1 000 000 cm³ a million, not a hundred 1 litre = 1 000 cm³ the capacity link 1 m³ = 1 000 litres so this follows

Volume multiplies three lengths, so the conversion factor is cubed.

So 1 m³ is 1 000 000 cm³. That figure surprises people, but it follows directly.

b) The link to capacity

1 litre = 1 000 cm³, and 1 millilitre = 1 cm³.

So 1 m³ holds 1 000 litres. A tank of 2.5 m³ holds 2 500 litres.

c) Volume of a cuboid and a cube

For a cuboid, volume = length × width × height.

For a cube, all edges are equal, so volume = side³.

All three measurements must be in the same unit before multiplying.

d) Volume of a cylinder

Volume of a cylinder Volume of a cylinder volume = π r² h base area times height volume = 22/7 × 7 × 7 × 20 radius 7 cm, height 20 cm volume = 3 080 cm³ the answer in litres = 3080 / 1000 divide by a thousand in litres = 3.08 litres its capacity

A cylinder's volume is the area of its circular base multiplied by its height.

Since the base area is πr², the volume is πr²h.

Use the radius, not the diameter.

e) Prisms in general

A prism has the same cross-section all the way along.

For any prism, volume = area of cross-section × length.

A cuboid is a prism with a rectangular cross-section. A cylinder is a prism with a circular one. So all three formulas are really the same rule.

For a triangular prism, find the area of the triangular end, then multiply by the length.

f) Solving problems

Read carefully whether the question wants volume or capacity, and answer in the units asked for.

A common question gives a tank's dimensions and asks how many litres it holds. Work out the volume in cm³, then divide by 1 000.

Another gives a rate of flow and asks how long to fill a tank. Find the volume, then divide by the rate.

g) Where this is used

Sizing a water tank. Ordering concrete by the cubic metre. Filling a drum. Costing packaging by volume.

Words to know

  • Volume -- the amount of space a solid occupies, in cubic units.
  • Capacity -- the amount a container can hold, in litres.
  • Prism -- a solid with the same cross-section throughout its length.
  • Cross-section -- the shape revealed by cutting a solid at right angles to its length.
  • Displacement -- the volume of liquid pushed aside by a submerged solid.

:::checkpoint Check yourself

  1. How many cm³ are in 1 m³?
  2. Find the volume of a cylinder with radius 14 cm and height 10 cm, taking π as 22/7.
  3. How many litres does a tank of 3 m³ hold?
  4. Why is the conversion factor cubed for volume units? :::

Bridge to practice

The exercises begin with units and conversion, move through cubes, cuboids, cylinders and the prism rule, and finish with capacity, composite solids and rate problems. In every applied question, convert to the unit the rate is expressed in before dividing.

Check yourselfPractise Volume and Capacity — 10 questions →Next in MathematicsMass, Weight and Density →