What you'll learn
Area is the amount of flat surface a shape covers, measured in square units such as square centimetres (cm²) or square metres (m²). If you covered a table top completely with 1 cm squares and counted them, that count is the area. This topic covers counting squares, the rectangle and square rules, the rule for a triangle, splitting an L-shaped figure into rectangles, working backwards when the area is given, and telling area apart from perimeter — which is the single biggest cause of lost marks in Measurement. Measurement carries 9 of the 40 items on the SEA Mathematics paper, and area is one of the topics that comes up almost every year, usually dressed up as tiles, paint, cloth or a garden plot.
Key terms and definitions
Area — the amount of surface a flat shape covers.
Square centimetre (cm²) — the area of a square that is 1 cm long and 1 cm wide.
Square metre (m²) — the area of a square that is 1 m long and 1 m wide. One square metre is 10 000 square centimetres.
Length and width — the two measurements of a rectangle. They are sometimes called length and breadth.
Base and height of a triangle — the side you rest the triangle on, and the straight-up distance from that side to the opposite corner.
Compound shape — a shape made from two or more rectangles joined together, such as an L-shaped room.
Perimeter — the distance around the edge. It is not area, and it is measured in plain cm or m.
Core concepts
Area is counting squares
Imagine a rectangle 5 cm long and 3 cm wide, ruled into 1 cm squares. There are 5 squares in each row and 3 rows, so there are 15 squares altogether. The area is 15 cm².
That is where the rule comes from, and it is worth saying properly: you multiply because you are counting rows of squares, not because someone told you to multiply.
Area of a rectangle = length × width.
Area of a square = side × side, because the length and the width are the same. A square of side 7 cm has area 7 × 7 = 49 cm².
Why the unit is squared
The answer counts squares, so the unit must be a square unit. Write cm², m² or km², and say it as "square centimetres". An area answer written as plain cm is marked wrong even when the number is right.
This also explains a conversion children often get wrong. One metre is 100 cm, so one square metre is 100 cm by 100 cm, which is 100 × 100 = 10 000 cm², not 100 cm². Squares grow much faster than lengths.
Area and perimeter are different questions
A rectangle 6 m by 2 m and a rectangle 4 m by 4 m are both quite different shapes, yet look at what they give:
| Rectangle | Perimeter | Area |
|---|---|---|
| 6 m by 2 m | (6 + 2) × 2 = 16 m | 6 × 2 = 12 m² |
| 4 m by 4 m | (4 + 4) × 2 = 16 m | 4 × 4 = 16 m² |
The perimeters are the same but the areas are not. Two shapes can have the same perimeter and different areas, and the other way round too. So you must decide which one the question is asking for before you calculate anything.
The clue is in the words. Fencing, ribbon, edging, running around, a border means perimeter. Tiles, paint, carpet, grass, cloth, covering means area.
Triangles
A triangle is exactly half of a rectangle drawn around it, so:
Area of a triangle = (base × height) ÷ 2.
A triangle with base 10 cm and height 6 cm has area (10 × 6) ÷ 2 = 60 ÷ 2 = 30 cm².
The height must be the straight-up distance from the base to the opposite corner, measured at a right angle to the base. It is not the slanted side, even though the slanted side is longer and more tempting.
Compound (L-shaped) figures
An L-shape is two rectangles joined, so cut it into two rectangles, find each area, and add. You can also treat it as a big rectangle with a corner removed and subtract. Both work, and getting the same answer twice is a good check.
Take an L-shaped verandah described by walking around it: 8 m right, 3 m up, 3 m left, 2 m up, 5 m left, 5 m down, back to the start.
Splitting method. The bottom strip is the full 8 m wide and 3 m tall: 8 × 3 = 24 m². The upper part sits above it, 5 m wide and 2 m tall: 5 × 2 = 10 m². Total 24 + 10 = 34 m².
Subtracting method. The whole rectangle around the shape is 8 m by 5 m, which is 40 m². The piece cut out of the top right corner is 3 m by 2 m, which is 6 m². So 40 − 6 = 34 m². The two methods agree.
Working backwards from a given area
If the area and one side are known, divide to find the other side.
A rectangular garden has area 48 m² and a width of 6 m. The length is 48 ÷ 6 = 8 m.
Once both sides are known you can go on to find the perimeter if the question asks: (8 + 6) × 2 = 28 m. Questions that ask for the perimeter after giving you the area are common, and they catch anyone who stops as soon as they find a number.
Worked examples
Example 1: Tiling a floor
Anisa's kitchen floor is a rectangle 6 m long and 4 m wide. The tiles are squares of side 50 cm. Tiles cost TT$18 each. How many tiles are needed, and what will they cost?
Work in the same unit. The tiles are 50 cm, which is 0.5 m.
Along the length: 6 ÷ 0.5 = 12 tiles. Across the width: 4 ÷ 0.5 = 8 tiles.
Number of tiles: 12 × 8 = 96 tiles.
Cost: 96 × 18. Split it — 96 × 10 = 960 and 96 × 8 = 768, so 960 + 768 = TT$1 728.
You can check with areas. The floor is 6 × 4 = 24 m². Each tile is 0.5 × 0.5 = 0.25 m². Then 24 ÷ 0.25 = 96 tiles, which matches.
Example 2: An L-shaped classroom carpet
A classroom is L-shaped. Its sides, taken in order around the edge, are 9 m, 4 m, 4 m, 3 m, 5 m and 7 m. Carpet costs TT$120 per square metre. How much will it cost to carpet the room?
Split it into two rectangles. The lower part is the full 9 m wide and 4 m tall: 9 × 4 = 36 m². The remaining part is 5 m wide and 3 m tall: 5 × 3 = 15 m². Total area = 36 + 15 = 51 m².
Check by subtraction. The bounding rectangle is 9 m by 7 m = 63 m², and the missing corner is 4 m by 3 m = 12 m². Then 63 − 12 = 51 m². The answers agree.
Cost: 51 × 120 = 51 × 12 × 10 = 612 × 10 = TT$6 120.
Example 3: From area back to perimeter
A rectangular banner has an area of 36 m². Its width is 4 m. How long is the banner, and how much braid is needed to trim all the way around its edge?
Length: 36 ÷ 4 = 9 m.
Braid goes around the edge, so this second part is a perimeter question: (9 + 4) × 2 = 13 × 2 = 26 m of braid.
The word "around" is the switch from area to perimeter. A pupil who answers 36 m has read the first half of the question only.
Common mistakes and how to avoid them
Adding the sides instead of multiplying. Answering 6 + 4 = 10 for a 6 m by 4 m room. Fix: ask whether the question is about covering (multiply) or going around (add).
Leaving off the square unit. Writing 24 m instead of 24 m². Fix: if you multiplied two lengths, the unit must be squared.
Thinking 1 m² = 100 cm². Fix: a square metre is 100 cm by 100 cm, so it is 10 000 cm².
Forgetting to halve for a triangle. Fix: a triangle is half its rectangle, so the final step is always ÷ 2.
Using the slanted side as the height of a triangle. Fix: the height goes straight up from the base at a right angle.
Mixing units in one calculation. Multiplying 6 m by 50 cm. Fix: change both to metres or both to centimetres first.
Splitting an L-shape and forgetting one piece. Fix: check with the subtraction method; if the two answers disagree, one piece is missing.
How parents can help at home
Area is easier to see than to explain, and your home is full of it.
Count tiles. Kitchen and bathroom floors are usually tiled in a grid. Ask your child to count the tiles along one wall and along the other, and then to work out the total without counting every tile. That is the area rule, discovered rather than memorised.
Ask the "cover it or go around it" question. Whenever a measurement comes up — a tablecloth, a fence, a tin of paint, a length of ribbon — ask, "is this a covering job or a going-around job?" Sorting that out is more than half of what the exam is testing.
Use the newspaper. Adverts for tiles, paint and carpet give a price per square metre. Give your child a room size and ask for the cost. Real prices in TT dollars make the practice feel worth doing.
When your child is stuck on an odd shape, the prompt is "cut it into rectangles and draw a line where you cut." Then ask for the two answers separately before adding them. If they cannot see the cut, ask instead, "what would the whole rectangle be if the corner were filled in?"
You do not need to check the arithmetic yourself. Ask your child to explain the answer back to you in words, ending with the unit. If the explanation makes sense and the unit is squared, it is very probably right.
Exam technique for area
Decide first whether the item is area or perimeter, and underline the word that tells you: tiles, paint and carpet mean area; fence, ribbon and border mean perimeter.
Convert to a single unit before multiplying. In tile questions, metres are usually easier than centimetres because the numbers stay small.
Show the two-step working when there is a cost. Write the area on one line and the cost on the next. If the second step goes wrong, clear working still shows the first step was right.
For a compound shape, do it both ways when you have time. Splitting and subtracting should agree, and that is the fastest self-check there is.
Check your unit at the end. Areas end in m² or cm²; costs end in TT$; a number of tiles has no unit at all, and it must be a whole number.
Quick revision summary
- Area is the number of unit squares a shape covers.
- Rectangle: area = length × width. Square: area = side × side.
- Triangle: area = (base × height) ÷ 2, using the straight-up height.
- Area units are squared: cm², m². 1 m² = 10 000 cm².
- Perimeter goes around the edge; area covers the inside. Read the question for which.
- L-shapes: split into rectangles and add, or take the whole rectangle and subtract the missing corner.
- Given the area and one side, divide to find the other side.
- Change everything to one unit before you multiply.