Mensuration means measuring. In this chapter you will measure the boundaries of flat shapes, the space they cover, and the room inside boxes. Before any of that, you need to be sure of the units: millimetres and kilometres, square centimetres and hectares. This lesson shows how the units are related, and why the units of area grow so much faster than the units of length.
Units of length
Length tells us how long something is. The main units are:
| Unit | Short form | Equal to |
|---|---|---|
| 1 centimetre | 1 cm | 10 mm |
| 1 metre | 1 m | 100 cm |
| 1 kilometre | 1 km | 1000 m |
So 1 m = 1000 mm, and 1 km = 1,00,000 cm.
To change a bigger unit into a smaller one, multiply. To change a smaller unit into a bigger one, divide. A bigger unit needs fewer of itself, so the number gets smaller when you go up.
Example 1
Change 3.5 km into metres, and 2 m 35 cm into centimetres.
1 km = 1000 m, so 3.5 km is m.
2 m is cm. So 2 m 35 cm is cm.
Example 2
Change 4750 m into kilometres, and 85 mm into centimetres.
, so 4750 m is 4.75 km.
, so 85 mm is 8.5 cm.
What area is
Area is the amount of flat surface a shape covers.
We measure area by counting squares. A square with sides of 1 cm has an area of 1 square centimetre, written 1 cm². A square with sides of 1 m has an area of 1 square metre, 1 m².
If a shape can be covered exactly by 12 squares of side 1 cm, its area is 12 cm².
Units of area
Here is the important point. A square of side 1 cm is also a square of side 10 mm. It can be cut into 10 rows of 10 small squares, each 1 mm by 1 mm. That is small squares. So
1 cm² = 100 mm², not 10 mm².
In the same way, a square of side 1 m is 100 cm by 100 cm, so
1 m² = cm² = 10,000 cm².
When a length unit is multiplied by 10, the area unit is multiplied by . Always multiply by the length factor twice.
For land, two bigger units are used:
| Unit | Equal to |
|---|---|
| 1 cm² | 100 mm² |
| 1 m² | 10,000 cm² |
| 1 hectare | 10,000 m² (a square of side 100 m) |
| 1 km² | 100 hectares = 10,00,000 m² |
A square of side 1 km is 1000 m by 1000 m, which is m². That is why 1 km² is 10,00,000 m².
Example 3
Change 4.2 m² into square centimetres.
1 m² = 10,000 cm², so 4.2 m² is cm².
Example 4
A farm has an area of 25,000 m². How many hectares is that?
1 hectare = 10,000 m², so the farm is hectares.
Example 5
Change 650 mm² into square centimetres.
1 cm² = 100 mm², so 650 mm² is cm².
Using the same unit
Before you add, subtract or multiply lengths, put them all in the same unit.
Example 6
A ribbon 2 m long has 75 cm cut off. How much is left?
2 m = 200 cm. Left cm, which is 1.25 m.
Example 7
A rectangular card is 1.2 m long and 50 cm wide. Find its area in square centimetres.
First write both lengths in cm: 1.2 m = 120 cm. The card can be covered by 120 columns of 50 squares, each of 1 cm². So its area is cm². This rule, area = length × breadth, is studied properly in a later lesson.
In square metres that is m². You can check: .
A common mistake
A student says: "1 m = 100 cm, so 1 m² = 100 cm²." This is wrong. A square metre is 100 cm along one side and 100 cm along the other. It holds squares of 1 cm². Remember: for area, use the length factor twice.
Practice
- Change 7.25 km into metres.
- Change 3 m 8 cm into centimetres.
- Change 680 cm into metres.
- Change 9 cm 4 mm into millimetres.
- Change 3 m² into square centimetres.
- Change 45,000 cm² into square metres.
- A plot of land is 3.6 hectares. How many square metres is it?
- Change 2.5 cm² into square millimetres.
- A rope is 5 m long. Pieces of 35 cm and 1 m 40 cm are cut from it. How much is left?
- How many hectares are there in 2 km²?
Answers
Show answers
- , so 7250 m.
- , so 308 cm.
- , so 6.8 m.
- , so 94 mm.
- , so 30,000 cm².
- , so 4.5 m².
- , so 36,000 m².
- , so 250 mm².
- Cut off cm. Left cm, which is 3.25 m.
- hectares.