Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Unit 1, lesson 2 (powers, squares and roots): what is 12²?
Answer
144. 12 × 12 = 144. - From Unit 1, lesson 6 (order of operations): what is 6 + 3 × 4?
Answer
18. Multiply first: 3 × 4 = 12, then 6 + 12 = 18. - From Unit 3, lesson 6 (solving simple equations): solve 4x = 52.
Answer
x = 13. Divide both sides by 4: 52 ÷ 4 = 13.
Words used in this lesson
- perimeter
- the distance all the way round the edge of a shape. For example: A fence round a field.
- area
- the amount of flat surface inside a shape, counted in unit squares. For example: The ground a lawn covers.
- square metre (m²)
- the area of a square 1 m by 1 m. For example: A floor of 12 m² holds 12 of them.
- length, width
- the two different side lengths of a rectangle. For example: A door 2 m long and 1 m wide in a drawing.
- rectangle
- a shape with four straight sides and four right angles. For example: A page, a door, a phone screen.
- scale factor
- the number every length is multiplied by. For example: All sides times 3 has scale factor 3.
Two different questions about one plot
Ngozi is setting up a vegetable plot in Enugu. She needs fence round the edge to keep the goats out, and compost to spread over the ground. They look like one question about size, but they are two: how far is it round, and how much ground is there to cover?
- Perimeter is the distance all the way round the edge of a shape. It is a length, so it is measured in metres (m), centimetres (cm), kilometres (km).
- Area is the amount of flat surface inside a shape. It is counted in squares of a fixed size: a square metre (m²) is a square 1 m by 1 m.
Predict first: make the rectangle one square longer and keep the width. Will the perimeter and the area both go up by the same amount? Write down a guess, then slide.
The rectangle is 6 m long and 4 m wide. Inside are 4 rows of 6 squares, so the area is 24 m². The distance round is 6 + 4 + 6 + 4 = 20 m.
Show me the steps
- At the opening size the rectangle is 6 m by 4 m. It has 4 rows of 6 squares, so the area is 24 m². The distance round is 6 + 4 + 6 + 4 = 20 m.
- Make the length 7 m and keep the width 4 m. The perimeter goes from 20 m to 22 m (one more metre at the top and one at the bottom), and the area goes from 24 m² to 28 m² (one new column of 4 squares). They do not grow by the same amount, because one counts a distance and the other counts squares.
- Make the width 1 m. There is one row of 6 squares, so the area is 6 m², but the perimeter is still 6 + 1 + 6 + 1 = 14 m: the edge goes round both sides of that thin strip.
- Make the length 10 m and the width 7 m. Seven rows of 10 squares give 70 m², and the perimeter is 10 + 7 + 10 + 7 = 34 m.
- Set both to 5 m. It is a square: 5 rows of 5 squares, 25 m², and 5 + 5 + 5 + 5 = 20 m round.
Perimeter is a distance; area is a count of squares.
- Perimeter of a rectangle: add the four sides, l + w + l + w, which is 2 × (l + w). Measured in m, cm or km.
- Area of a rectangle: l × w. Measured in m², cm² or km².
Why l × w works: the rectangle is w rows with l squares in each row, and w rows of l squares is w × l squares. The units tell you which is which: a distance in m, a count of squares in m². It still works when a side is not whole: 2.5 m by 4 m is 4 rows of two and a half squares, 4 × 2.5 = 10 m². Back to Ngozi: the fence is the perimeter, in metres, and the compost is the area, in square metres. Two numbers, two units.
A common mistake: “The plot is 13 m by 4 m, so its area is 13 + 4 = 17 m².” Adding the two sides gives half the perimeter, not the area. Count the rows: 4 rows of 13 squares is 13 × 4 = 52 m². Before you write a unit, ask whether the number is a distance (m) or a count of squares (m²).
When every side grows
A designer enlarges a poster so that every length is multiplied by the same number. The edge gets longer by that number. But the area has two directions, along and up, and both grow. Multiply every side and see how many copies of the first rectangle fit inside.
Every side is 2 times as long: 8 m by 6 m. The perimeter is 28 m, 2 times the first 14 m. The area is 48 m², 2 × 2 = 4 times the first 12 m², because 4 copies of the first rectangle fit inside.
Show me the steps
- The first rectangle is 4 m by 3 m. Its perimeter is 14 m and its area is 12 m².
- Multiply every side by 2. The rectangle is 8 m by 6 m: perimeter 28 m, which is 2 × 14, and area 48 m², which is 4 × 12. Four copies of the first rectangle fit.
- Multiply every side by 3. It is 12 m by 9 m: perimeter 42 m (3 × 14) and area 108 m² (9 × 12). Nine copies fit, three rows of three.
- Multiply every side by 4. It is 16 m by 12 m: perimeter 56 m (4 × 14) and area 192 m² (16 × 12). Sixteen copies fit.
Doubling every side doubles the perimeter but makes the area four times as big: two rows of two copies. Multiplying every side by k multiplies the perimeter by k and the area by k × k.
The same rule explains a common unit slip. A square metre is a square 100 cm by 100 cm, so it holds 100 × 100 = 10 000 cm², not 100. A length in centimetres is 100 times the length in metres, so an area in square centimetres is 100 × 100 times the area in square metres.
Worked example: an ordinary rectangle
A courtyard is 11 m long and 6 m wide. Find its perimeter and its area.
- Perimeter: add the four sides, 11 + 6 + 11 + 6 = 34 m. (Or 2 × (11 + 6) = 2 × 17 = 34 m.)
- Area: 6 rows of 11 squares, 11 × 6 = 66.
- Units: the perimeter is a distance, 34 m. The area counts square metres, 66 m².
Your go. The first step is given. An allotment is 14 m long and 8 m wide. Find its perimeter and its area.
- Perimeter: add the four sides, 14 + 8 + 14 + 8.
- Now find the area yourself, then decide on each unit, then check.
Check the full answer
Perimeter 14 + 8 + 14 + 8 = 44 m. Area 14 × 8 = 112 m². The perimeter is a distance in m and the area is a count of squares in m².
Check before you go on
1. A rectangle is 8 cm long and 3 cm wide. Which statement is correct?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A rectangular field is 15 m long and 8 m wide. What is its area in square metres?
3. A picture frame is 32.5 cm long and 20 cm wide. How far is it round the outside edge, in centimetres?
Level 2 · Building
4. The floor of a rectangular classroom has an area of 72 m² and is 9 m long. How wide is it, in metres?
5. A rectangular plot has a perimeter of 38 m and is 12 m long. How wide is it, in metres?
6. Every side of a rectangle is made 5 times as long. How many times as big is its area?
Level 3 · Stretch
7. A rectangular garden is 14 m long and has an area of 98 m². A fence goes round all four sides. How many metres of fence are needed?
8. A square has a perimeter of 36 m. What is its area in square metres?
Something to think about
A rectangle has perimeter 20 m. With whole-number sides it can have an area of 9 m². Could a rectangle with this perimeter have an area of 30 m²? Try some widths and look for the biggest area you can make. Then: a 1 m by 1 m square is also 100 cm by 100 cm, so how many square centimetres is 1 m²?
One way to think about it
Widths 1, 2, 3, 4 and 5 give lengths 9, 8, 7, 6 and 5, so areas 9, 16, 21, 24 and 25. After that the pairs repeat in the other order. The most is 25 m², the square, so 30 m² is not possible. Lesson 2 asks why the square wins. And 1 m² = 100 × 100 = 10 000 cm², not 100 cm².
Exit ticket
9. Hana says: “My bed in the school garden has an area of 12 m², so I need 12 m of edging.” What is wrong with that?