Pick one before you read on. It is fine to be wrong.
By the end you can
Find the area of a rectangle as length × width
Find the area of an L shape by splitting it
Explain why 1 m² is 10 000 cm²
You want to cover a floor with square tiles. How many tiles do you need? The space inside a shape is its area. We measure area by counting squares.
A square centimetre is a square that is 1 cm long and 1 cm wide. We write it as 1 cm². An area of 24 cm² means 24 of these squares fit inside the shape.
Draw it first
Count the squares in rows
1 Draw the rectangle on squared paper. Each square is 1 cm by 1 cm.
2 One row has 6 squares. There are 4 rows.
3 Count all the squares with a sum: 6 × 4 = 24 cm².
Rows of equal size make a short cut. The number of squares in a row is the length. The number of rows is the width. So the area of a rectangle is its length × its width.
Worked example
A floor is a rectangle 7 m long and 3 m wide. What is its area?
Imagine squares that are 1 m by 1 m. Each is a square metre, written 1 m².
There are 3 rows with 7 squares in each row.
7 × 3 = 21 m².
Answer: 21 m²
Each square is 1 m by 1 m. There are 3 rows of 7 squares.
A square is a rectangle with equal sides. A square with sides of 6 m has an area of 6 × 6 = 36 m². Always write the unit with a small 2, such as cm² or m², because you are counting squares.
Some shapes are not rectangles. An L shape is made of two rectangles joined together. To find its area, split it into rectangles, find each area, and add them.
Worked example
Find the area of this L shape by splitting it with a flat line.
Draw a dashed line to split the L into two rectangles.
The bottom rectangle is 8 m by 3 m: 8 × 3 = 24 m².
The top rectangle is 4 m by 3 m: 4 × 3 = 12 m².
Add the two areas: 24 + 12 = 36 m².
Answer: 36 m²
Split with a flat line: 24 + 12 = 36 m².
You can split the L another way, with an upright line. The two parts change, but the total does not.
Split with an upright line: 4 × 6 = 24 and 4 × 3 = 12. The total is 36 m² again.
There is a third way. Fill in the missing corner to make one big rectangle. Find its area, then take the corner away.
Worked example
Find the area of the same L shape by filling in the corner.
Draw the rectangle that would hold the L. It is 8 m by 6 m.
Its area is 8 × 6 = 48 m².
The corner that is missing is 4 m by 3 m: 4 × 3 = 12 m².
Take the corner away: 48 − 12 = 36 m².
Answer: 36 m²
The whole rectangle is 8 × 6 = 48 m². The dotted corner is 4 m by 3 m, so 4 × 3 = 12 m². The L is 48 − 12 = 36 m².
Now think small. A square metre is a square that is 1 m long and 1 m wide. But 1 m is also 100 cm. So a square metre is a square 100 cm long and 100 cm wide.
The square metre. Each small square is 10 cm by 10 cm.One small square from the picture above, made bigger. It holds 10 × 10 = 100 squares of 1 cm².
Worked example
Why is 1 m² the same as 10 000 cm²?
1 m is 100 cm.
A square metre is 100 cm long and 100 cm wide.
Cut it into squares of 10 cm by 10 cm. You get 10 across and 10 down, which is 100 squares.
Each small square is 10 × 10 = 100 cm².
100 squares of 100 cm² each make 100 × 100 = 10 000 cm².
Answer:10 000 cm²
So 1 m² is not 100 cm². It is 100 times 100, which is 10 000 cm². Area counts squares. A square has a length and a width, so the 100 is used twice.
See it three ways
With things
On squared paper, make 3 rows with 5 squares in each row. Count all the squares.
As a picture
In symbols
Area = length × width = 5 × 3 = 15 cm².
Try it
Count the squares, then multiply
You need: squared paper with squares 1 cm by 1 cm, a pencil, a ruler.
Draw a rectangle that is 7 squares long and 5 squares wide.
Count the squares one by one. Write the number.
Now multiply 7 × 5. Compare the two answers.
What you will see: Counting and multiplying give the same area, 35 cm². Multiplying is faster, and it works for any size.
If you cannot try it, this is what happens.
A wrong idea many people have
A shape with a longer perimeter must have a bigger area.
Why it feels right: A longer edge seems to hold more.
The test that breaks it: Draw a strip 8 cm by 1 cm and a square 3 cm by 3 cm. The strip has the longer perimeter, 18 cm against 12 cm. The square has the bigger area, 9 cm² against 8 cm².
What is true: Perimeter is the distance round. Area is the space inside. They are two different measures, and one does not tell you the other.
Key words
area
the space inside a shape, counted in squares.
square centimetre
a square 1 cm long and 1 cm wide. Written cm².
square metre
a square 1 m long and 1 m wide. Written m².
Back to your guess
The answer is C: Multiply the length by the width of the floor.
Tiles cover the space inside the floor. A floor 7 tiles long and 3 tiles wide has 3 rows of 7 tiles. That is 7 × 3 = 21 tiles. So you multiply the length by the width.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
What is the area of this rectangle?
See the answer
45 cm²
The area is length × width: 9 × 5 = 45 cm².
If you chose B: You added 9 + 5. Area counts squares in rows, so you multiply the length by the width. Go to M3.6.3.
If you chose C: You added all four sides. That is the perimeter, the distance round. The area is length × width. Go to M3.6.2.
A square has sides of 6 m. What is its area?
See the answer
36 m²
A square is a rectangle with equal sides. The area is 6 × 6 = 36 m².
If you chose A: You added 6 + 6. Multiply the two sides instead: the length and the width are both 6 m. Go to M3.6.3.
If you chose C: You added all four sides, 4 × 6. That is the perimeter. The area is 6 × 6. Go to M3.6.2.
What is the area of this L shape?
See the answer
23 m²
Split the L with a flat line. The bottom rectangle is 7 × 2 = 14 m². The top rectangle is 3 × 3 = 9 m². Then 14 + 9 = 23 m².
If you chose B: You found the area of the whole rectangle, 7 × 5. The L has a corner missing. Take the corner away, or split the L into two rectangles. Go to M3.6.3.
If you chose C: You added all six sides. That is the perimeter. The area is the space inside. Go to M3.6.2.
How many square centimetres are in one square metre?
See the answer
10 000 cm²
1 m is 100 cm. A square 100 cm by 100 cm has an area of 100 × 100 = 10 000 cm².
If you chose A: You used the length, 100 cm, and stopped. A square metre is 100 cm long and 100 cm wide, so multiply 100 × 100. Go to M3.6.3.
If you chose B: That is one zero too few. 100 × 100 has four zeros. Go to M3.6.3.
What is one square centimetre?
See the answer
A square with sides one centimetre long
Area is measured in squares. One square centimetre is the area of a square with sides one centimetre long.
If you chose A: A line one centimetre long is just a centimetre. A square centimetre is a square, so it has space inside. Go to M3.6.3.
If you chose C: A cube measures how much space a solid takes up. That is volume, which comes later. A square centimetre is flat. Go to M3.6.3.
A rectangle has an area of 40 cm². Its length is 8 cm. How wide is it?
See the answer
5 cm
Length × width = 40, so the width is 40 ÷ 8 = 5 cm. Check: 8 × 5 = 40.
If you chose A: You took 8 from 40. Area is length × width, so go back with a division. Go to M3.6.3.
If you chose B: You multiplied 40 × 8. The area is already the product, so divide it by the length instead. Go to M3.6.3.
A rectangle is 9 cm long and 4 cm wide. What is its area?
See the answer
36 cm²
The area is length × width: 9 × 4 = 36 cm².
From an earlier chapter
A rectangle is 12 m long and 5 m wide. What is its perimeter?
See the answer
34 m
Add the four sides: 12 + 5 + 12 + 5 = 34 m.
Keep it
Copy this into your exercise book.
Area is the space inside, counted in squares. Rectangle: length × width. L shape: split it, or fill and take away. 1 m² = 100 cm × 100 cm = 10 000 cm².
Draw: A rectangle on squared paper, 6 by 4, with its squares drawn. Write 6 × 4 = 24 cm² beside it.
