GCSE Maths: Units of area and area in context (Higher)
Part of the GCSE Maths perimeter, area and circles chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Units of area and area in context
Area units are square units, so converting them is different from converting lengths. Here is the key picture. A square that is 1 m by 1 m has an area of 1 m². But 1 m is 100 cm, so the same square measures 100 cm by 100 cm. Its area is 100 × 100 = 10 000 cm². So
1 m² = 10 000 cm² (not 100!)
In the same way 1 cm = 10 mm, so 1 cm² = 10 × 10 = 100 mm², and 1 km = 1000 m, so 1 km² = 1 000 000 m². The rule: square the length conversion. Going from a big unit to a small unit gives a bigger number (multiply); going from small to big gives a smaller number (divide).
Area problems in context
Real problems use area to decide how much to buy and what it costs. Follow the same plan every time: (1) change all lengths to the same unit, (2) find the area, (3) divide by the amount one unit covers (one tile, one tin, one roll), (4) round up to a whole number of items, (5) multiply by the price.
Why round up? If you need 7.2 tins you cannot buy 7.2 tins, and 7 tins would not be enough. Use perimeter for things that go round the edge (fences, borders, edging strip) and area for things that cover the surface (paint, turf, tiles, carpet).
Key ideas
- 1 m² = 100 × 100 = 10 000 cm²; 1 cm² = 10 × 10 = 100 mm²; 1 km² = 1 000 000 m².
- Area conversions: square the length conversion. Big unit to small unit: multiply. Small to big: divide.
- Make all lengths the same unit before finding an area.
- Items that cover a surface (paint, tiles, turf) use area. Items round the edge (fencing, edging) use perimeter.
- If you need part of an item, round up to a whole item, then multiply by the price.
Watch out
To change m² to cm² multiply by 10 000, not by 100. A quick check: a 1 m by 1 m square is 100 cm by 100 cm, and 100 × 100 = 10 000.Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 100 cm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in cm it is 100 cm by 100 cm, which is 100 × 100 small squares.1 m² = 100 × 100 = 10,000 cm²
- Convert: m² to cm² gives a bigger number. Each m² is 10,000 cm², so multiply.3.5 × 10,000 = 35,000 cm²
Check. Go back: 35,000 ÷ 10,000 = 3.5 m². ✓
Watch out. 1 m = 100 cm, but 1 m² = 10,000 cm², not 100. Areas need the conversion squared.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 100 cm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in cm it is 100 cm by 100 cm, which is 100 × 100 small squares.1 m² = 100 × 100 = 10,000 cm²
- Convert: cm² to m² gives a smaller number. It takes 10,000 cm² to make one m², so divide.450,000 ÷ 10,000 = 45 m²
Check. Go back: 45 × 10,000 = 450,000 cm². ✓
Watch out. 1 m = 100 cm, but 1 m² = 10,000 cm², not 100. Areas need the conversion squared.
Method. Change both lengths to the same unit first, then multiply. Here we change centimetres to metres.
- Change 80 cm to metres. There are 100 cm in 1 m, so divide by 100.80 ÷ 100 = 0.8 m
- Multiply the lengths. Both are now in metres, so the answer is in m².2.5 × 0.8 = 2
(25 × 8 = 200, two decimal places, so 2.00) - Write the units. Metres times metres gives square metres.Area = 2 m²
Check. In cm: 250 × 80 = 20 000 cm², and 20 000 ÷ 10 000 = 2 m². ✓
Watch out. Never multiply 2.5 m by 80 cm directly: change one so the units match.
Method. Find how many tiles fit along each side (in the same unit), then multiply. Or divide the floor area by the area of one tile.
- Change the floor to centimetres. The tile is in cm, so use cm for everything.3.2 m = 320 cm and 2.4 m = 240 cm
- Tiles along the length. Divide the length by the tile side.320 ÷ 40 = 8
- Tiles along the width. Divide the width by the tile side.240 ÷ 40 = 6
- Total tiles. 8 columns of 6 tiles.8 × 6 = 48 tiles
Check. Areas: floor 320 × 240 = 76 800 cm²; tile 40 × 40 = 1600 cm²; 76 800 ÷ 1600 = 48. ✓
Watch out. Do not mix metres and centimetres in the same division.
Method. Find the area, double it for two coats, divide by the cover per litre, work out how many tins, and multiply by the price.
- Area of the wall. length × height.5 × 2.4 = 12 m²
- Area to paint with two coats. Each coat covers the whole wall.12 × 2 = 24 m²
- Litres needed. One litre covers 8 m².24 ÷ 8 = 3 litres
- Tins needed. Each tin holds 2.5 litres. 3 litres is more than one tin, so find 3 ÷ 2.5 and round up.3 ÷ 2.5 = 1.2, so 2 tins
- Total cost. Multiply by the price of one tin.2 × 18.50 = £37
Check. 2 tins hold 5 litres, enough for the 3 litres needed; 1 tin (2.5 litres) is not enough. ✓
Watch out. Round up the number of tins. 1.2 tins means you must buy 2.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 1000 mm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in mm it is 1000 mm by 1000 mm, which is 1000 × 1000 small squares.1 m² = 1000 × 1000 = 1,000,000 mm²
- Convert: mm² to m² gives a smaller number. It takes 1,000,000 mm² to make one m², so divide.3,600,000 ÷ 1,000,000 = 3.6 m²
Check. Go back: 3.6 × 1,000,000 = 3,600,000 mm². ✓
Watch out. 1 m = 1000 mm, but 1 m² = 1,000,000 mm², not 1000. Areas need the conversion squared.
Exercise 18E Units of area and area in context
Calculator allowedMethod. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 100 cm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in cm it is 100 cm by 100 cm, which is 100 × 100 small squares.1 m² = 100 × 100 = 10,000 cm²
- Convert: m² to cm² gives a bigger number. Each m² is 10,000 cm², so multiply.3 × 10,000 = 30,000 cm²
Check. Go back: 30,000 ÷ 10,000 = 3 m². ✓
Watch out. 1 m = 100 cm, but 1 m² = 10,000 cm², not 100. Areas need the conversion squared.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 100 cm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in cm it is 100 cm by 100 cm, which is 100 × 100 small squares.1 m² = 100 × 100 = 10,000 cm²
- Convert: m² to cm² gives a bigger number. Each m² is 10,000 cm², so multiply.0.5 × 10,000 = 5000 cm²
Check. Go back: 5000 ÷ 10,000 = 0.5 m². ✓
Watch out. 1 m = 100 cm, but 1 m² = 10,000 cm², not 100. Areas need the conversion squared.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 cm = 10 mm
- Square it to compare the areas. 1 cm² is a square 1 cm by 1 cm. Measured in mm it is 10 mm by 10 mm, which is 10 × 10 small squares.1 cm² = 10 × 10 = 100 mm²
- Convert: cm² to mm² gives a bigger number. Each cm² is 100 mm², so multiply.7 × 100 = 700 mm²
Check. Go back: 700 ÷ 100 = 7 cm². ✓
Watch out. 1 cm = 10 mm, but 1 cm² = 100 mm², not 10. Areas need the conversion squared.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 km = 1000 m
- Square it to compare the areas. 1 km² is a square 1 km by 1 km. Measured in m it is 1000 m by 1000 m, which is 1000 × 1000 small squares.1 km² = 1000 × 1000 = 1,000,000 m²
- Convert: km² to m² gives a bigger number. Each km² is 1,000,000 m², so multiply.8 × 1,000,000 = 8,000,000 m²
Check. Go back: 8,000,000 ÷ 1,000,000 = 8 km². ✓
Watch out. 1 km = 1000 m, but 1 km² = 1,000,000 m², not 1000. Areas need the conversion squared.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 100 cm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in cm it is 100 cm by 100 cm, which is 100 × 100 small squares.1 m² = 100 × 100 = 10,000 cm²
- Convert: cm² to m² gives a smaller number. It takes 10,000 cm² to make one m², so divide.45,000 ÷ 10,000 = 4.5 m²
Check. Go back: 4.5 × 10,000 = 45,000 cm². ✓
Watch out. 1 m = 100 cm, but 1 m² = 10,000 cm², not 100. Areas need the conversion squared.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 100 cm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in cm it is 100 cm by 100 cm, which is 100 × 100 small squares.1 m² = 100 × 100 = 10,000 cm²
- Convert: cm² to m² gives a smaller number. It takes 10,000 cm² to make one m², so divide.250,000 ÷ 10,000 = 25 m²
Check. Go back: 25 × 10,000 = 250,000 cm². ✓
Watch out. 1 m = 100 cm, but 1 m² = 10,000 cm², not 100. Areas need the conversion squared.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 cm = 10 mm
- Square it to compare the areas. 1 cm² is a square 1 cm by 1 cm. Measured in mm it is 10 mm by 10 mm, which is 10 × 10 small squares.1 cm² = 10 × 10 = 100 mm²
- Convert: mm² to cm² gives a smaller number. It takes 100 mm² to make one cm², so divide.600 ÷ 100 = 6 cm²
Check. Go back: 6 × 100 = 600 mm². ✓
Watch out. 1 cm = 10 mm, but 1 cm² = 100 mm², not 10. Areas need the conversion squared.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 km = 1000 m
- Square it to compare the areas. 1 km² is a square 1 km by 1 km. Measured in m it is 1000 m by 1000 m, which is 1000 × 1000 small squares.1 km² = 1000 × 1000 = 1,000,000 m²
- Convert: m² to km² gives a smaller number. It takes 1,000,000 m² to make one km², so divide.3,500,000 ÷ 1,000,000 = 3.5 km²
Check. Go back: 3.5 × 1,000,000 = 3,500,000 m². ✓
Watch out. 1 km = 1000 m, but 1 km² = 1,000,000 m², not 1000. Areas need the conversion squared.
Method. Change 250 cm to metres so that both lengths are in the same unit, then multiply.
- Change 250 cm to metres. Divide by 100.250 ÷ 100 = 2.5 m
- Multiply the lengths. Both are in metres, so the area is in m².4 × 2.5 = 10 m²
- Write the units. Area is in square metres.Area = 10 m²
Check. In cm: 400 × 250 = 100 000 cm² and 100 000 ÷ 10 000 = 10 m². ✓
Watch out. Make the units match before you multiply.
Method. Change 1.5 m to centimetres, then multiply.
- Change to centimetres. 1 m = 100 cm, so multiply by 100.1.5 × 100 = 150 cm
- Multiply the lengths. Both are in cm, so the area is in cm².150 × 60 = 9000 cm²
Check. In metres: 1.5 × 0.6 = 0.9 m² and 0.9 × 10 000 = 9000 cm². ✓
Watch out. 1 m² is 10 000 cm², not 100.
Method. Find how many tiles are needed, then multiply by the price of one tile.
- Change to centimetres. The tile is measured in cm.3.2 m = 320 cm, 2.4 m = 240 cm
- Tiles along each side. Divide by 40.320 ÷ 40 = 8 and 240 ÷ 40 = 6
- Number of tiles. 8 × 6.8 × 6 = 48
- Cost. Each tile costs £2.75.48 × 2.75 = £132.00
(48 × 275 = 13 200, two decimal places)
Check. Estimate: 50 × £3 = £150, close to £132. ✓
Watch out. Divide lengths in the same unit.
Method. Find the area, double it for two coats, find the litres, round up the tins, then multiply by the price.
- Area of the wall. width × height.6 × 2.5 = 15 m²
- Two coats. Each coat covers the whole wall.15 × 2 = 30 m²
- Litres needed. One litre covers 10 m².30 ÷ 10 = 3 litres
- Tins needed. Each tin is 2 litres, so divide and round up.3 ÷ 2 = 1.5, so 2 tins
- Cost. Each tin costs £14.40.2 × 14.40 = £28.80
Check. 2 tins = 4 litres, which covers 40 m²; that is at least 30 m². ✓
Watch out. 1.5 tins means 2 tins: round up.
Method. Find the area of the lawn, divide by the area of one roll, and multiply by the price.
- Area of the lawn. length × width.9 × 6 = 54 m²
- Number of rolls. Each roll covers 1.5 m².54 ÷ 1.5 = 36 rolls
(540 ÷ 15 = 36) - Cost. Each roll costs £3.20.36 × 3.20 = £115.20
Check. 36 rolls cover 36 × 1.5 = 54 m². ✓
Watch out. Use area for turf: it covers the surface.
Method. Find the perimeter, take off the gate gap, then multiply by the cost per metre.
- Perimeter of the garden. Add all four sides.14 + 9 + 14 + 9 = 46 m
- Take away the gate gap. No fence is needed there.46 − 3 = 43 m
- Cost. Each metre costs £18.43 × 18 = £774
(43 × 10 = 430, 43 × 8 = 344, 430 + 344 = 774)
Check. Estimate: 45 × 18 ≈ 810, close to 774. ✓
Watch out. Fencing is a perimeter job. Remember to subtract the gap.
Method. Find the area, work out how many of each tin you need, and compare the total costs.
- Area of the wall. length × height.9 × 5 = 45 m²
- Tins of A. Each covers 30 m². 45 ÷ 30 = 1.5, so round up.2 tins of A: 2 × £19 = £38
- Tins of B. Each covers 60 m², which is more than 45 m².1 tin of B = £34
- Compare. Subtract.£38 − £34 = £4
Check. Two tins of A cover 60 m², the same as one tin of B, so the prices compare fairly. ✓
Watch out. Round up the number of tins before comparing costs.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 1000 mm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in mm it is 1000 mm by 1000 mm, which is 1000 × 1000 small squares.1 m² = 1000 × 1000 = 1,000,000 mm²
- Convert: mm² to m² gives a smaller number. It takes 1,000,000 mm² to make one m², so divide.3,600,000 ÷ 1,000,000 = 3.6 m²
Check. Go back: 3.6 × 1,000,000 = 3,600,000 mm². ✓
Watch out. 1 m = 1000 mm, but 1 m² = 1,000,000 mm², not 1000. Areas need the conversion squared.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 km = 1000 m
- Square it to compare the areas. 1 km² is a square 1 km by 1 km. Measured in m it is 1000 m by 1000 m, which is 1000 × 1000 small squares.1 km² = 1000 × 1000 = 1,000,000 m²
- Convert: km² to m² gives a bigger number. Each km² is 1,000,000 m², so multiply.0.0025 × 1,000,000 = 2500 m²
Check. Go back: 2500 ÷ 1,000,000 = 0.0025 km². ✓
Watch out. 1 km = 1000 m, but 1 km² = 1,000,000 m², not 1000. Areas need the conversion squared.
Method. Change the field to m², then divide by the size of one plot.
- Change km² into m². 1 km² = 1 000 000 m².0.045 × 1 000 000 = 45 000 m²
- Divide into plots. Each plot is 450 m².45 000 ÷ 450 = 100
- Remainder. 100 × 450 = 45 000, which uses the whole field.45 000 − 45 000 = 0 m²
Check. 100 × 450 = 45 000 m² = 0.045 km². ✓
Watch out. Square the length conversion: 1 km = 1000 m, so 1 km² = 1 000 000 m².
Method. Work out the area of a 1 m by 1 m square twice: once in metres, once in centimetres.
- Area in metres. Side 1 m.1 × 1 = 1 m²
- Area in centimetres. The same square has sides of 100 cm.100 × 100 = 10 000 cm²
- Conclude. It is the same square, so the areas are equal.1 m² = 10 000 cm², not 100 cm². Ben only converted one of the two lengths.
Check. A 1 m wide strip with 100 rows of 100 squares of 1 cm: 100 × 100 = 10 000. ✓
Watch out. An area has two lengths multiplied, so the conversion factor is used twice.
Method. Work out the area in cm² and convert, then check by working in metres.
- Area in cm². length × width.60 × 40 = 2400 cm²
- Convert to m². 1 m² = 10 000 cm², so divide by 10 000.2400 ÷ 10 000 = 0.24 m²
- Check in metres. 60 cm = 0.6 m and 40 cm = 0.4 m.0.6 × 0.4 = 0.24 m²
- Conclude. Yes, Mia is correct.
Check. Both routes give 0.24 m². ✓
Watch out. Divide by 10 000, not by 100, when changing cm² into m².
Method. Find the side from the area, change it to centimetres, and work out the perimeter.
- Find the side. The side is the square root of the area. 15 × 15 = 225, so 1.5 × 1.5 = 2.25.√2.25 = 1.5 m
- Change to centimetres. 1 m = 100 cm.1.5 × 100 = 150 cm
- Perimeter. Four equal sides.4 × 150 = 600 cm
- Conclude. The perimeter is 600 cm, as required.
Check. 4 × 1.5 = 6 m = 600 cm. ✓
Watch out. The square root of the area gives the side, not the perimeter.