Shapes, Space and Data
Primary 5 · Mathematics
Shapes, Space and Data
Lesson 2 of 3
15 min
Let's begin
Habimana's mother is making a kitchen garden behind the house. It is a rectangle, 12 m long and 7 m wide.
She has two questions. First: how many metres of wire will I need to fence right round it? Second: how much manure must I buy to cover the ground inside it?
Same garden — but these are two completely different measurements, and mixing them up costs real money. Today you will learn to tell them apart, and to add a third: how much a box can hold.
Imagine walking all the way round the edge of the garden, back to where you started. The total distance you walk is the perimeter.
Let us just walk it. The garden is 12 m long and 7 m wide, so its four sides are 12 m, 7 m, 12 m and 7 m. Add all four:
12 + 7 + 12 + 7 = 38 m.
So Habimana's mother needs 38 m of wire.
Now look at what we added: two 12s and two 7s. Instead of writing four numbers we can add one long side to one short side, then double it:
(12 + 7) = 19, and 19 × 2 = 38 m. Same answer, less writing.
Perimeter of a rectangle = 2 × (length + width)
A square is easier still, because all four sides are the same:
Perimeter of a square = 4 × side
Perimeter is a length, so it is measured in ordinary length units: cm, m, km.
Wire costs 500 Frw a metre. What will it cost to fence right round the 12 m by 7 m garden?
Step 1 Find the perimeter: 2 × (12 + 7) = 2 × 19 = 38 m.
Step 2 Multiply by the price of one metre: 38 × 500 = 19,000 Frw.
Now for the manure. Manure does not go round the edge — it covers the whole flat inside. The amount of flat surface a shape covers is called its area.
Do this on squared paper. Draw a rectangle 5 squares long and 3 squares wide, where each little square is 1 cm across. Now count the squares inside.
You get 3 rows, and each row has 5 squares in it. So 5 + 5 + 5 = 15 squares — or much faster, 5 × 3 = 15 squares.
Each of those little squares is 1 centimetre by 1 centimetre. We call one of them a square centimetre and write it cm².
So the area of that rectangle is 15 cm².
Notice what you actually did: you multiplied the length by the width. That is all area ever is.
Area of a rectangle = length × width
Area of a square = side × side
Area is measured in square units: cm², m², km². The little raised 2 is your reminder that you are counting squares.
So the manure must cover 12 × 7 = 84 m² of ground.
Draw a rectangle 8 cm long and 5 cm wide. Its area is 8 × 5 = 40 cm².
Now cut it in half along the diagonal, from one corner straight across to the opposite corner. You are holding two triangles — and they are exactly the same size. If you turn one over it fits perfectly on top of the other.
So each triangle must be half of the rectangle: 40 ÷ 2 = 20 cm².
Look at one of those triangles. Its bottom edge is 8 cm — we call that the base. It stands 5 cm tall — we call that the height. And its area came out as half of 8 × 5.
Area of a triangle = 12 × base × height
Be careful about the height. It is the straight-up distance from the base to the top point, measured at a right angle to the base. It is not the slanting side. If a question gives you a slanting side as well, that number is there to catch you out — ignore it.
A triangular flag has a base of 10 cm and a height of 6 cm. What is its area?
Step 1 Multiply base by height: 10 × 6 = 60.
Step 2 Take half of it: 60 ÷ 2 = 30 cm².
(If you had forgotten the half, you would have said 60 cm² — which is the whole rectangle, twice as much flag as you really have.)
Perimeter goes round a shape. Area covers the inside of a flat shape. But a box is not flat — you can pour beans into it. The amount of space inside a solid is its volume.
Picture a small wooden box, 5 cm long, 3 cm wide and 4 cm high. Now imagine filling it with tiny cubes that are 1 cm on every edge.
Layer 1. Cover the bottom of the box with cubes. You fit 5 along and 3 across, so the bottom layer holds 5 × 3 = 15 cubes.
Now stack up. The box is 4 cm high, and each cube is 1 cm high, so you can fit 4 layers.
Total cubes: 15 × 4 = 60 cubes.
Each cube is one cubic centimetre, written cm³. So the volume of the box is 60 cm³.
Look at what you multiplied: 5 × 3 × 4.
Volume of a cuboid = length × width × height
Volume is measured in cubic units: cm³, m³. The little raised 3 reminds you that you are counting cubes.
Niyonzima has a carton 8 cm long, 5 cm wide and 4 cm high. What is its volume?
Step 1 Bottom layer: 8 × 5 = 40 cubes.
Step 2 There are 4 layers: 40 × 4 = 160 cm³.
You could also just write it in one line: 8 × 5 × 4 = 160 cm³.
A rectangle has an area of 72 cm². Its length is 9 cm. What is its perimeter?
Step 1 Area = length × width, so 9 × width = 72. That means width = 72 ÷ 9 = 8 cm.
Step 2 Now that you know both sides, find the perimeter: 2 × (9 + 8) = 2 × 17 = 34 cm.
The trick was to use the area to find the missing side first. Never try to jump straight to the answer.
| If the job is… | You need… | Units look like… |
|---|---|---|
| Fencing, a rope, a border, a frame | Perimeter | m, cm |
| Tiles, paint, grass, manure, cloth | Area | m², cm² |
| Water, sand, beans inside a box | Volume | m³, cm³ |
Tip
Round it, cover it, fill it
And let the units check your work. Count how many lengths you multiplied together:
one length → m • two lengths → m² • three lengths → m³
If your answer to an area question does not end in a little 2, stop and look again.
Do each one in your book before you look. Every answer is worked out for you underneath.
Now measure your own exercise book with a ruler. Find its perimeter in cm, then its area in cm².
What you learned today
Quick check
A garden is 9 m long and 5 m wide. How much wire do you need to fence right round it?
