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Primary Maths · Year 6Chapter 9 · Measures, area and volume
Year 6 · Chapter 9

Year 6 Area, Volume and Measures: worked examples and questions with answers

By the end of this chapter you will be able to

  • Change between metric units with decimals up to three places, moving the decimal point the right way
  • Convert between hours, minutes and seconds, including decimal hours
  • Convert between miles and kilometres, using 1 mile is about 1.6 km
  • Find the area of a parallelogram and a triangle, using the perpendicular height and not the slanted side
  • Show that shapes with the same area can have different perimeters, and that a cut corner keeps the perimeter the same
  • Say when a formula can be used for the area or volume of a shape, and when a shape must be split up first
  • Find and compare the volume of cubes and cuboids, and convert between cm³, m³, mm³ and km³

You should already know

  • Multiplying and dividing decimals by 10, 100 and 1000 (Year 6, Chapter 5)
  • Area of rectangles and perimeter, and the formula P = 2a + 2b (Year 5, Chapter 8; Year 6, Chapter 8)
  • Multiplying and dividing whole numbers and decimals (Year 6, Chapters 2 and 5)
  • Equivalent units of length, mass, capacity and time (Year 5, Chapter 8)

Where this leads

  • Shapes, angles and circles, including the circumference and area of a circle (Year 6, Chapter 10)
  • Using formulae and rearranging them, and the volume of prisms (KS3)
  • Ratio and scale factors for lengths, areas and volumes (KS3)

National Curriculum for England, Key Stage 2 mathematics, Year 6. Content: Solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal places, Use, read, write and convert between standard units (length, mass, volume, time) from a smaller unit to a larger unit and vice versa, Convert between miles and kilometres, Recognise that shapes with the same areas can have different perimeters and vice versa, Recognise when it is possible to use formulae for area and volume of shapes, Calculate the area of parallelograms and triangles, Calculate, estimate and compare the volume of cubes and cuboids using cm³ and m³, and extend to mm³ and km³.

Exercise P  Do you remember?

Non-calculator
1
FWork out 3.6 × 10.
2
FWork out 4.2 × 100.
3
FA rectangle is 7 cm long and 5 cm wide. What is its area, in cm²?
4
FA rectangle has a = 8 cm and b = 3 cm. Use the formula P = 2a + 2b to find its perimeter, in cm.
5
FWork out 2.5 × 4.
9.1

Changing metric units with decimals

Year 6
Use, read, write and convert between standard units, from a smaller unit to a larger unit and vice versa, Solve problems involving the conversion of units, using decimal notation up to three decimal places
Video lesson: Changing metric units with decimalsComing soon

The metric units are linked by powers of ten. A kilometre is 1000 metres, a metre is 100 centimetres, and a metre is 1000 millimetres. Grams and kilograms work the same way, and so do litres and millilitres.

Going to a smaller unit makes the number bigger, so we multiply. Going to a bigger unit makes the number smaller, so we divide. Each power of ten moves the decimal point one place, so 1000 moves it three places.

km× 1000m×100cm×10mm
Going down the ladder to smaller units multiplies. Going up to bigger units divides.

Key ideas

  • 1 km = 1000 m, 1 m = 100 cm, 1 m = 1000 mm, 1 cm = 10 mm.
  • 1 kg = 1000 g, and 1 litre = 1000 ml.
  • Smaller unit: the number gets bigger, so multiply.
  • Bigger unit: the number gets smaller, so divide.
  • Decimals work the same way: 0.725 litres = 725 ml, because 0.725 × 1000 = 725.
Worked example 9.1.1
Change 2.375 kilometres to metres.

Key words. To convert means to change a measure into another unit. Kilo means 1000, so 1 km = 1000 m. Centi means one hundredth, so 1 m = 100 cm. Milli means one thousandth, so 1 m = 1000 mm.

Method. Find how many m make one km, then multiply by that number.

  1. Find how many m make one km. The table says 1 km = 1000 m.
    1 km = 1000 m
  2. Multiply by 1000. The answer is in m, which is smaller than km, so there are more m. Multiplying makes the number bigger.
    2.375 × 1000 = 2375
  3. Write the answer. Moving the decimal point 3 places to the right gives the answer in m.
    2375

Check. 2375 ÷ 1000 = 2.375, the number we started with. ✓

Watch out. A smaller unit gives a bigger number, so multiply. Move the decimal point to the right.

Worked example 9.1.2
Change 4650 grams to kilograms.

Key words. To convert means to change a measure into another unit. Kilo means 1000, so 1 km = 1000 m. Centi means one hundredth, so 1 m = 100 cm. Milli means one thousandth, so 1 m = 1000 mm.

Method. Find how many g make one kg, then divide by that number.

  1. Find how many g make one kg. The table says 1 kg = 1000 g.
    1 kg = 1000 g
  2. Divide by 1000. The answer is in kg, which is bigger than g, so there are fewer kg. Dividing makes the number smaller.
    4650 ÷ 1000 = 4.65
  3. Write the answer. Moving the decimal point 3 places to the left gives the answer in kg.
    4.65

Check. 4.65 × 1000 = 4650, the number we started with. ✓

Watch out. A bigger unit gives a smaller number, so divide. Move the decimal point to the left.

Worked example 9.1.3
Change 36.5 centimetres to millimetres.

Key words. To convert means to change a measure into another unit. Kilo means 1000, so 1 km = 1000 m. Centi means one hundredth, so 1 m = 100 cm. Milli means one thousandth, so 1 m = 1000 mm.

Method. Find how many mm make one cm, then multiply by that number.

  1. Find how many mm make one cm. The table says 1 cm = 10 mm.
    1 cm = 10 mm
  2. Multiply by 10. The answer is in mm, which is smaller than cm, so there are more mm. Multiplying makes the number bigger.
    36.5 × 10 = 365
  3. Write the answer. Moving the decimal point 1 place to the right gives the answer in mm.
    365

Check. 365 ÷ 10 = 36.5, the number we started with. ✓

Watch out. A smaller unit gives a bigger number, so multiply. Move the decimal point to the right.

Watch out

Common mistake: 1 km is 1000 m, not 100 m. Check the direction. Metres are a smaller unit than kilometres, so 2.375 km gives a bigger number of metres. Grams are smaller than kilograms, so 4650 g gives a smaller number of kilograms.

Exam tip

Count the steps between the two units on the ladder, then move the decimal point that many places.

Exercise 9A  Changing metric units

Non-calculator
1
FChange 3.2 kilometres to metres.
2
FChange 5.75 metres to centimetres.
3
FChange 0.6 kilograms to grams.
4
FChange 0.725 litres to millilitres.
5
FChange 480 millimetres to centimetres.
6
FChange 750 grams to kilograms.
7
FChange 6.04 metres to millimetres.
8
FChange 1450 millilitres to litres.
9
PSA ribbon is 2.45 m long. Ali cuts off 85 cm. How many centimetres of ribbon are left?
10
PSA bag of flour has 1.25 kg in it. Mia uses 375 g. How many grams of flour are left?
11
PSA bottle holds 0.75 litres of juice. How many 50 ml cups can be filled from the bottle, if none is spilled?
12
RSam says that 0.4 km is more than 450 m. Is Sam right? Explain your answer.
9.2

Converting units of time

Year 6
Use, read, write and convert between standard units of time, from a smaller unit to a larger unit and vice versa, Solve problems involving the conversion of units of time, using decimal notation
Video lesson: Converting units of timeComing soon

Time is measured in hours, minutes and seconds. There are 60 minutes in 1 hour and 60 seconds in 1 minute. Sometimes we write part of an hour as a decimal, so 30 minutes is 0.5 hours and 45 minutes is 0.75 hours.

To change hours to minutes, multiply by 60. To change minutes to hours, divide by 60. To change hours and minutes to a decimal, change the minutes to a fraction of an hour first, then add the whole hours.

00.250.50.7511.251.51.7522.252.52.7532 h 15 min = 2.25 h
The number line shows each quarter of an hour as a decimal: 0.25, 0.5, 0.75 and so on.

Key ideas

  • 1 hour = 60 minutes, so 1 hour = 60 min.
  • 1 minute = 60 seconds.
  • 0.5 hours = 30 minutes, 0.25 hours = 15 minutes, 0.75 hours = 45 minutes.
  • 2 hours 15 minutes = 2.25 hours, because 15 minutes is 0.25 of an hour.
Worked example 9.2.1
A film lasts 1.75 hours. How many minutes is this?

Key words. There are 60 minutes in 1 hour and 60 seconds in 1 minute. A decimal hour uses a decimal point: 30 minutes is 0.5 hours, because 30 is half of 60.

Method. Multiply the hours by 60 to get minutes.

  1. Remember: 1 hour = 60 minutes. So each hour gives 60 minutes.
    1 h = 60 min
  2. Multiply the hours by 60. This gives the number of minutes.
    1.75 × 60 = 105
  3. Write the answer. The answer is in minutes.
    105

Check. 105 ÷ 60 = 1.75 hours. ✓

Watch out. Multiply by 60 to go from hours to minutes. Do not multiply by 100.

Worked example 9.2.2
Change 45 minutes into hours. Write your answer as a decimal.

Key words. There are 60 minutes in 1 hour and 60 seconds in 1 minute. A decimal hour uses a decimal point: 30 minutes is 0.5 hours, because 30 is half of 60.

Method. Divide the minutes by 60 to get hours.

  1. Remember: 60 minutes = 1 hour. So we divide the minutes by 60.
    60 min = 1 h
  2. Divide the minutes by 60. This gives the number of hours, written as a decimal.
    45 ÷ 60 = 0.75
  3. Write the answer. The answer is in hours.
    0.75

Check. 0.75 × 60 = 45 minutes. ✓

Watch out. Divide by 60 to go from minutes to hours. For example, 30 minutes is half of an hour, so it is 0.5 hours.

Worked example 9.2.3
Change 2 hours 15 minutes into hours. Write your answer as a decimal.

Key words. There are 60 minutes in 1 hour and 60 seconds in 1 minute. A decimal hour uses a decimal point: 30 minutes is 0.5 hours, because 30 is half of 60.

Method. Change the minutes to hours with a decimal, then add the whole hours.

  1. Change the minutes to a fraction of an hour. Each minute is one sixtieth of an hour.
    15 ÷ 60 = 0.25
  2. Add the whole hours. The whole hours stay the same.
    2 + 0.25 = 2.25
  3. Write the answer. The answer is in hours, as a decimal.
    2.25

Check. 2.25 hours is 2 hours and 15 minutes: 15 ÷ 60 = 0.25. ✓

Watch out. Do not write 2 h 15 min as 2.15. The 15 minutes is a quarter of an hour, so it is 0.25.

Watch out

Common mistake: 2 hours 15 minutes is not 2.15 hours. The 15 minutes is a quarter of an hour, which is 0.25, so the answer is 2.25 hours.

Exam tip

Ask yourself: how many 60s are in the time? A decimal of an hour is a part of 60 minutes, so 0.5 is 30 minutes.

Exercise 9B  Converting time

Non-calculator
1
FChange 2.5 hours into minutes.
2
FChange 3.25 hours into minutes.
3
FChange 0.4 hours into minutes.
4
FChange 2.5 minutes into seconds.
5
FChange 30 minutes into hours. Write your answer as a decimal.
6
FChange 27 minutes into hours. Write your answer as a decimal.
7
FChange 1 hour 30 minutes into hours. Write your answer as a decimal.
8
FChange 3 hours 45 minutes into hours. Write your answer as a decimal.
9
PSA pan of soup simmers for 1.25 hours. How many minutes does it simmer for?
10
PSA train journey takes 2.5 hours. A bus journey takes 165 minutes. How many minutes longer is the bus journey?
11
PSA runner finishes a race in 3 minutes 20 seconds. How many seconds is this?
12
RTom says that 0.6 hours is the same as 60 minutes. Is Tom right? Explain your answer.
9.3

Miles and kilometres

Year 6
Convert between miles and kilometres, Solve problems involving measures, using decimal notation up to three decimal places
Video lesson: Miles and kilometresComing soon

Miles and kilometres are both used for distance. We use the approximate rule 1 mile is about 1.6 kilometres. The sign ≈ means "is about equal to", so the answers are approximate equivalents.

To change miles to kilometres, multiply by 1.6. To change kilometres to miles, divide by 1.6. It helps to check that a bigger distance in miles gives a bigger number of kilometres.

Miles51020
Kilometres (about)81632
Each mile is about 1.6 kilometres, so 5 miles is about 8 kilometres.

Key ideas

  • 1 mile is about 1.6 km.
  • Miles to kilometres: multiply by 1.6.
  • Kilometres to miles: divide by 1.6.
  • A kilometre is shorter than a mile, so the number of kilometres is bigger than the number of miles.
Worked example 9.3.1
Change 15 miles to kilometres. The answer is approximate.

Key words. 1 mile is about 1.6 kilometres (1 mile ≈ 1.6 km). The sign ≈ means "is about equal to".

Method. Multiply the miles by 1.6, because 1 mile is about 1.6 km.

  1. Use the rule: 1 mile is about 1.6 km. So we multiply the miles by 1.6 to get kilometres.
    1 mile ≈ 1.6 km
  2. Multiply the miles by 1.6. It is easier to multiply by 16 and then divide by 10.
    15 × 16 = 240, then 240 ÷ 10 = 24
  3. Write the answer. The answer is in kilometres.
    24

Check. 24 ÷ 1.6 = 15 miles. ✓

Watch out. Miles to kilometres: multiply. Kilometres to miles: divide. Do not divide when you go from miles to kilometres.

Worked example 9.3.2
Change 12 kilometres to miles.

Key words. 1 mile is about 1.6 kilometres (1 mile ≈ 1.6 km). The sign ≈ means "is about equal to".

Method. Divide the kilometres by 1.6, because 1 mile is about 1.6 km.

  1. Use the rule: 1 mile is about 1.6 km. So we divide the kilometres by 1.6 to get miles.
    1 mile ≈ 1.6 km
  2. Divide by 1.6. To avoid a decimal divisor, multiply the top and bottom by 10. Then 1.6 becomes 16.
    12 ÷ 1.6 = 120 ÷ 16
  3. Work out the division. The answer may have a decimal part, which must end.
    120 ÷ 16 = 7.5
  4. Write the answer. The answer is in miles.
    7.5

Check. 7.5 × 1.6 = 12 km. ✓

Watch out. Kilometres to miles: divide by 1.6. A mile is longer than a kilometre, so the number of miles is smaller.

Worked example 9.3.3
A cycle path is 25 kilometres long. How many miles is this? Give the exact answer, to three decimal places if needed.

Key words. 1 mile is about 1.6 kilometres (1 mile ≈ 1.6 km). The sign ≈ means "is about equal to".

Method. Divide the kilometres by 1.6, because 1 mile is about 1.6 km.

  1. Use the rule: 1 mile is about 1.6 km. So we divide the kilometres by 1.6 to get miles.
    1 mile ≈ 1.6 km
  2. Divide by 1.6. To avoid a decimal divisor, multiply the top and bottom by 10. Then 1.6 becomes 16.
    25 ÷ 1.6 = 250 ÷ 16
  3. Work out the division. The answer may have a decimal part, which must end.
    250 ÷ 16 = 15.625
  4. Write the answer. The answer is in miles.
    15.625

Check. 15.625 × 1.6 = 25 km. ✓

Watch out. Kilometres to miles: divide by 1.6. A mile is longer than a kilometre, so the number of miles is smaller.

Watch out

Common mistake: 1 mile is more than 1 kilometre. A mile is about 1.6 km, so when you change kilometres to miles, the answer must be less than the number of kilometres. Divide, do not multiply.

Exam tip

Check your answer by changing it back. For example, 7.5 miles × 1.6 should give 12 km.

Exercise 9C  Miles and kilometres

Non-calculator
1
FChange 10 miles to kilometres.
2
FChange 20 miles to kilometres.
3
FChange 5 miles to kilometres.
4
FChange 8 kilometres to miles.
5
FChange 16 kilometres to miles.
6
FChange 3 kilometres to miles. Give your answer as an exact decimal.
7
FChange 2.5 miles to kilometres.
8
FChange 24 kilometres to miles.
9
PSA school trip is 12 miles long. About how many kilometres is this?
10
PSJo runs 10 kilometres. Her friend runs 6 miles. How many more miles than her friend does Jo run?
11
PSA charity walk is 26 miles long. How many kilometres is this?
12
RSam says that 1 mile is about 2 km, so 10 miles is about 20 km. Is Sam right? Explain your answer.
9.4

Area of parallelograms and triangles

Year 6
Calculate the area of parallelograms and triangles, Recognise when it is possible to use a formula for the area of a shape
Video lesson: Area of parallelograms and trianglesComing soon

A parallelogram is a four-sided shape with two pairs of parallel sides. Its area is the base times the perpendicular height. The height is measured straight up from the base, at a right angle to it.

A parallelogram can be cut along its height and moved to make a rectangle with the same base and height. So the area is base × height. The slanted side is not used.

A triangle is half of a parallelogram with the same base and height. So its area is half of base × height, or (base × height) ÷ 2.

12 cm8 cm10 cm
A parallelogram: the base is 12 cm and the perpendicular height is 8 cm. The slanted side is 10 cm, but it is not used.
10 cm7 cm
A triangle: the base is 10 cm and the perpendicular height is 7 cm.

Key ideas

  • Parallelogram: area = base × perpendicular height.
  • Triangle: area = (base × perpendicular height) ÷ 2.
  • Use the perpendicular height, not the slanted side.
  • Area is measured in square units, such as cm².
Worked example 9.4.1
A parallelogram has a base of 12 cm, a perpendicular height of 8 cm and a slanted side of 10 cm. Find its area, in cm².

Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.

Method. Multiply the base by the perpendicular height. Do not use the slanted side.

  1. Find the base and the perpendicular height. The height goes straight up from the base. The slanted side is not the height.
    base = 12 cm, perpendicular height = 8 cm (the slanted side, 10 cm, is not used)
  2. Multiply the base by the perpendicular height. The area of a parallelogram is base times perpendicular height.
    12 × 8 = 96
  3. Write the answer. The area is in square centimetres.
    96

Check. Cut the triangle off one end and move it to the other: a rectangle 12 cm by 8 cm has the same area, 96 cm². ✓

Watch out. Use the vertical height, not the slanted side. The slanted side is always longer than the height.

Worked example 9.4.2
A triangle has a base of 10 cm and a perpendicular height of 7 cm. Find its area, in cm².

Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.

Method. Multiply the base by the height, then divide by 2.

  1. Find the base and the perpendicular height. The height is the straight line from the top corner to the base, at a right angle.
    base = 10 cm, perpendicular height = 7 cm
  2. Multiply the base by the height. Two copies of the triangle make a parallelogram with the same base and height.
    10 × 7 = 70
  3. Halve the answer. One triangle is half of the parallelogram.
    70 ÷ 2 = 35
  4. Write the answer. The area is in square centimetres.
    35

Check. Two of these triangles make a parallelogram with area 70 cm², so one triangle is 70 ÷ 2 = 35. ✓

Watch out. Do not forget the divide by 2. A triangle is half of a rectangle with the same base and height.

Worked example 9.4.3
A triangle has a base of 13 cm and a perpendicular height of 6 cm. Find its area, in cm².

Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.

Method. Multiply the base by the height, then divide by 2.

  1. Find the base and the perpendicular height. The height is the straight line from the top corner to the base, at a right angle.
    base = 13 cm, perpendicular height = 6 cm
  2. Multiply the base by the height. Two copies of the triangle make a parallelogram with the same base and height.
    13 × 6 = 78
  3. Halve the answer. One triangle is half of the parallelogram.
    78 ÷ 2 = 39
  4. Write the answer. The area is in square centimetres.
    39

Check. Two of these triangles make a parallelogram with area 78 cm², so one triangle is 78 ÷ 2 = 39. ✓

Watch out. Do not forget the divide by 2. A triangle is half of a rectangle with the same base and height.

Watch out

Common mistake: using the slanted side as the height. The slanted side is always longer than the height. Also, for a triangle, do not forget to divide by 2.

Exam tip

A triangle is half of a rectangle with the same base and height. If the answer is bigger than the rectangle, check your working.

Exercise 9D  Areas of parallelograms and triangles

Non-calculator
1
FA parallelogram has a base of 6 cm and a perpendicular height of 4 cm. Find its area, in cm².
2
FA parallelogram has a base of 9 cm and a perpendicular height of 5 cm. Find its area, in cm².
3
FA parallelogram has a base of 12 cm and a perpendicular height of 7 cm. Find its area, in cm².
4
FA triangle has a base of 8 cm and a perpendicular height of 6 cm. Find its area, in cm².
5
FA triangle has a base of 5 cm and a perpendicular height of 4 cm. Find its area, in cm².
6
FA triangle has a base of 7 cm and a perpendicular height of 6 cm. Find its area, in cm².
7
FA triangle has a base of 9 cm and a perpendicular height of 5 cm. Find its area, in cm².
8
FA parallelogram has a base of 15 cm and a perpendicular height of 4 cm. Find its area, in cm².
9
PSLook at the parallelogram. Find its area, in cm². Do not use the slanted side.
10 cm12 cm13 cm
10
PSA triangle has a base of 14 cm and a perpendicular height of 9 cm. Find its area, in cm².
11
PSA parallelogram has an area of 48 cm² and a base of 8 cm. Find its perpendicular height, in cm.
12
RTwo triangles have the same base of 10 cm. Triangle A has a perpendicular height of 6 cm. Triangle B has a perpendicular height of 9 cm. How many times bigger is the area of triangle B than the area of triangle A?
9.5

Same area, different perimeters, and when to use a formula

Year 6
Recognise that shapes with the same areas can have different perimeters, and vice versa, Recognise when it is possible to use formulae for the area of a shape
Video lesson: Same area, different perimeters, and when to use a formulaComing soon

Two shapes can have the same area and different perimeters. Look at the rectangles with an area of 36 cm²: a long thin rectangle has a big perimeter, and the square has the smallest perimeter.

Length (cm)Width (cm)Area (cm²)Perimeter (cm)
1363674
2183640
3123630
493626
663624
Every rectangle in the table has an area of 36 cm², but the perimeters are different.

The reverse is also true: shapes with the same perimeter can have different areas. Cutting a corner out of a rectangle keeps the perimeter the same, but makes the area smaller.

A formula can be used for the standard shapes: a rectangle, a square, a parallelogram, a triangle and a cuboid. For any other shape, such as an L-shape, first split it into standard shapes, then use the formula for each part.

3 cm2 cm8 cm5 cm
A rectangle 8 cm by 5 cm with a 3 cm by 2 cm corner cut out. The dashed lines show the cut-out piece.

Key ideas

  • Shapes with the same area can have different perimeters.
  • Shapes with the same perimeter can have different areas.
  • Cutting a corner out of a rectangle keeps the perimeter the same and makes the area smaller.
  • Use a formula for a standard shape (rectangle, square, parallelogram, triangle, cuboid). For any other shape, split it into standard shapes first.
Worked example 9.5.1
Rectangles with whole-number sides each have an area of 24 cm². Which of them has the smallest perimeter? Give the perimeter in cm.

Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.

Method. List the rectangles with this area, work out each perimeter, and choose the smallest.

  1. Find every pair of whole-number sides that multiply to make the area. The sides must multiply to 24.
    1 × 24, 2 × 12, 3 × 8, 4 × 6
  2. Find the perimeter of each rectangle. Perimeter = 2 times (length + width).
    1 × 24: 2 × (1 + 24) = 50 ; 2 × 12: 2 × (2 + 12) = 28 ; 3 × 8: 2 × (3 + 8) = 22 ; 4 × 6: 2 × (4 + 6) = 20
  3. Choose the smallest perimeter. The smallest number in the list is the answer.
    The smallest is 20
  4. Write the answer. The perimeter is in centimetres.
    20

Check. Every pair was checked, so no rectangle is missed. ✓

Watch out. A square has the smallest perimeter for its area. Rectangles that are long and thin have big perimeters.

Worked example 9.5.2
A rectangle is 8 cm by 5 cm. A 3 cm by 2 cm corner is cut out. Find the area of the shape that is left, in cm².

Key words. The area is the space inside a flat shape, in square units (cm²). The perpendicular height is measured straight up from the base, at a right angle to it. It is not the slanted side.

Method. Find the area of the whole rectangle, then take away the piece that is cut out.

  1. Find the area of the whole rectangle. Area is length times width.
    8 × 5 = 40
  2. Find the area of the corner that is cut out, 3 cm by 2 cm. The cut-out piece is a rectangle too.
    3 × 2 = 6
  3. Take the cut-out area away. The shape that is left has the rest of the area.
    40 - 6 = 34
  4. Write the answer. The area is in square centimetres.
    34

Check. Add the two parts back: 6 + 34 = 40. ✓

Watch out. Do not add the two areas. The cut-out piece is taken away.

Worked example 9.5.3
A rectangle with whole-number sides has a perimeter of 24 cm. What is the biggest area it can have, in cm²?

Key words. The perimeter is the total distance all the way round the outside of a shape. Add the lengths of all the outside sides.

Method. Work out the area of every rectangle with this perimeter, and choose the largest.

  1. Find each pair of whole-number sides that makes this perimeter. Each pair must add up to 12.
    1 + 11 = 12, 2 + 10 = 12, 3 + 9 = 12, 4 + 8 = 12, 5 + 7 = 12, 6 + 6 = 12
  2. Find the area of each rectangle. Area is length times width.
    1 × 11 = 11 ; 2 × 10 = 20 ; 3 × 9 = 27 ; 4 × 8 = 32 ; 5 × 7 = 35 ; 6 × 6 = 36
  3. Choose the biggest area. The biggest number in the list is the answer.
    The biggest area is 36
  4. Write the answer. The area is in square centimetres.
    36

Check. The rectangle with sides 6 cm and 6 cm has area 6 × 6 = 36 cm². ✓

Watch out. Shapes with the same perimeter can have different areas. The most square shape has the biggest area.

Watch out

Common mistake: thinking that the same area means the same perimeter. A 1 cm by 36 cm strip has an area of 36 cm² but a perimeter of 74 cm. Always work out the perimeter from the sides.

Exam tip

When a shape is not a rectangle, split it into rectangles. Work out the area of each part and add them together.

Exercise 9E  Area, perimeter and formulae

Non-calculator
1
FA rectangle is 9 cm long and 4 cm wide. What is its perimeter, in cm?
2
FA rectangle is 9 cm long and 4 cm wide. What is its area, in cm²?
3
FHow many different rectangles have whole-number sides and an area of 36 cm²?
4
FA square has sides of 6 cm. What is its perimeter, in cm?
5
FA rectangle with whole-number sides has a perimeter of 20 cm. What is the biggest area it can have, in cm²?
6
FA rectangle is 10 cm by 6 cm. A 4 cm by 2 cm corner is cut out. What is the area of the shape that is left, in cm²?
7
FWhat is the perimeter of the same shape, in cm?
8
PSRectangle A is 5 cm by 6 cm. Rectangle B is 3 cm by 10 cm. Both have an area of 30 cm². What is the difference between their perimeters, in cm?
9
PSA rectangle is 5 cm by 7 cm. A square is 6 cm by 6 cm. Both shapes have a perimeter of 24 cm. What is the difference between their areas, in cm²?
10
PSThe picture shows a 7 cm by 5 cm rectangle with a 2 cm by 2 cm corner cut out. What is the area of the shape, in cm²?
2 cm2 cm7 cm5 cm
11
RAli says that cutting a corner out of a rectangle makes its perimeter smaller. Look at the shape in question 10. Is Ali right? Explain your answer.
12
RCould a rectangle with whole-number sides have an area of 36 cm² and a perimeter of 25 cm? Explain your answer.
9.6

Volume of cubes and cuboids

Year 6
Calculate, estimate and compare the volume of cubes and cuboids using cm³ and m³, and extend to other units (mm³, km³), Recognise when it is possible to use a formula for the volume of a shape
Video lesson: Volume of cubes and cuboidsComing soon

The volume is the amount of space a solid takes up. We measure it by counting cubes. A cuboid with length l, width w and height h has volume l × w × h. A cube has all its edges the same, so its volume is side × side × side.

A layer of 1 cm cubes that is 5 cubes long and 4 cubes wide has 5 × 4 = 20 cubes. If the cuboid is 3 layers high, the volume is 20 × 3 = 60 cm³.

The formula can be used for a cuboid or a cube. For a solid made of two or more cuboids, use the formula for each cuboid and add the volumes. The formula cannot be used on the whole solid if it is not a cuboid.

One layer of 1 cm cubes: 5 cubes long and 4 cubes wide, so 20 cubes.
LengthVolume
1 cm = 10 mm1 cm³ = 1000 mm³
1 m = 100 cm1 m³ = 1 000 000 cm³
1 km = 1000 m1 km³ = 1 000 000 000 m³
Cubic units use the length change three times over. For example, 1 m is 100 cm, so 1 m³ is 100 × 100 × 100 cm³.

Key ideas

  • Cuboid: volume = length × width × height.
  • Cube: volume = side × side × side.
  • Volume is measured in cubic units, such as cm³ or m³.
  • 1 m³ = 1 000 000 cm³ (100 × 100 × 100). 1 km³ = 1 000 000 000 m³.
6 cm3 cm4 cm
A cuboid 6 cm long, 4 cm wide and 3 cm high.
Worked example 9.6.1
A cuboid is 6 cm long, 4 cm wide and 3 cm high. Find its volume, in cm³.

Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.

Method. Multiply the length, the width and the height.

  1. Write the formula. Volume = length times width times height.
    V = l × w × h
  2. Multiply the length by the width. This gives the area of the base.
    6 × 4 = 24
  3. Multiply by the height. This gives the number of cubes that fill the box.
    24 × 3 = 72
  4. Write the answer. The volume is in cubic units.
    72

Check. Order does not matter: 3 × 4 × 6 = 72. ✓

Watch out. Volume is in cubic units, such as cm³. Multiply all three lengths in the same unit first.

Worked example 9.6.2
A cube has sides of 0.5 m. Find its volume, in m³.

Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.

Method. Multiply the side by itself three times.

  1. Write the formula. A cube has three equal edges, so its volume is side times side times side.
    V = s × s × s
  2. Multiply the first two sides. This is the area of one face.
    0.5 × 0.5 = 0.25
  3. Multiply by the third side. This gives the volume.
    0.25 × 0.5 = 0.125
  4. Write the answer. The volume is in cubic units.
    0.125

Check. Each face has area 0.25, and the cube is 0.5 deep, so the volume is 0.25 × 0.5 = 0.125. ✓

Watch out. Do not multiply the side by 3. Volume is side times side times side, which is much more than 3 times the side.

Worked example 9.6.3
A cuboid is 2.5 m long, 0.4 m wide and 1.2 m high. Find its volume in cm³.

Key words. The volume is the space inside a solid, in cubic units (cm³ or m³). A cuboid has six rectangular faces. A cube has six square faces, and all its edges are the same length.

Method. Find the volume in m³, then change it to cm³.

  1. Multiply the length by the width. Multiply the decimals first.
    2.5 × 0.4 = 1.0
  2. Multiply by the height. This gives the volume in m³.
    1.0 × 1.2 = 1.2 m³
  3. Change 1.2 m³ to cm³. 1 m³ = 1 000 000 cm³, so multiply by 1 000 000.
    1.2 × 1 000 000 = 1200000
  4. Write the answer. The volume is in cm³.
    1200000

Check. 1200000 ÷ 1000000 = 1.2 m³, the volume we found first. ✓

Watch out. A cubic metre is a million cubic centimetres. Do not multiply by 100 only.

Watch out

Common mistake: 1 m³ is not 100 cm³. It is 100 × 100 × 100 = 1 000 000 cm³. Cubic units use the length change three times over.

Exam tip

If an answer is in cubic units, and the question asks for a bigger unit, the number must get smaller: divide.

Exercise 9F  Volume of cubes and cuboids

Non-calculator
1
FA cuboid is 5 cm long, 2 cm wide and 4 cm high. What is its volume, in cm³?
2
FA cube has sides of 3 cm. What is its volume, in cm³?
3
FA cuboid is 1.5 cm long, 2 cm wide and 4 cm high. What is its volume, in cm³?
4
FA solid is made from two cuboids joined together. One cuboid is 4 cm by 3 cm by 2 cm. The other is 4 cm by 3 cm by 1 cm. What is the total volume, in cm³?
5
FHow many cubic centimetres are there in 1 m³?
6
FHow many cubic metres are there in 1 km³?
7
FChange 3000000 cm³ to m³.
8
FChange 4 cm³ to mm³.
9
PSHow many 1 cm cubes fit exactly into a box that is 12 cm long, 5 cm wide and 4 cm high?
10
PSCuboid A is 6 cm by 3 cm by 2 cm. Cuboid B is 4 cm by 4 cm by 3 cm. How many cubic centimetres more is cuboid B than cuboid A?
11
PSA water tank is 1.5 m long, 1 m wide and 0.8 m high. How many litres of water can it hold? (1 m³ holds 1000 litres.)
12
RSam says that if you double every edge of a cube, its volume doubles. Is Sam right? Explain your answer.

Chapter summary

SkillMethod
Changing unitsGoing to a smaller unit: multiply. Going to a bigger unit: divide. Move the decimal point one place for each power of ten.
Time1 hour = 60 minutes. Hours to minutes: multiply by 60. Minutes to hours: divide by 60. Put the part of an hour after the point.
Miles and kilometres1 mile is about 1.6 km. Miles to kilometres: multiply by 1.6. Kilometres to miles: divide by 1.6.
ParallelogramArea = base × perpendicular height. Do not use the slanted side.
TriangleArea = (base × perpendicular height) ÷ 2.
Perimeter and areaShapes with the same area can have different perimeters. Cutting a corner out keeps the perimeter the same.
VolumeCuboid: length × width × height. Cube: side × side × side. 1 m³ = 1 000 000 cm³.

Can you…?

  • I can change between metric units, including decimals up to three places.
  • I can convert between hours, minutes and seconds, including decimal hours.
  • I can convert between miles and kilometres, using 1 mile is about 1.6 km.
  • I can find the area of a parallelogram and a triangle, using the perpendicular height.
  • I can explain why shapes with the same area can have different perimeters.
  • I can say when a formula can be used for the area or volume of a shape.
  • I can find the volume of a cube or cuboid, and convert cubic units.

Key terms

Convert
To change a measure into another unit, for example metres into centimetres.
Perpendicular height
The straight height from the top of a shape to its base, at a right angle to the base.
Parallelogram
A four-sided shape with two pairs of parallel sides.
Area
The amount of space inside a flat shape, in square units.
Volume
The amount of space inside a solid, in cubic units.
Cuboid
A solid with six rectangular faces.
Cube
A solid with six square faces and all its edges the same length.
Perimeter
The total distance all the way round the outside of a shape.

Exercise 9Z  Check-up (non-calculator)

Non-calculator
1
Change 4.6 kilometres to metres.
2
Change 0.35 litres to millilitres.
3
Change 2.5 hours into minutes.
4
Change 1 hour 15 minutes into hours. Write your answer as a decimal.
5
Change 16 kilometres to miles.
6
A parallelogram has a base of 11 cm and a perpendicular height of 5 cm. Find its area, in cm².
7
A triangle has a base of 12 cm and a perpendicular height of 9 cm. Find its area, in cm².
8
A rectangle is 9 cm by 7 cm. A 3 cm by 2 cm corner is cut out. What is the perimeter of the shape, in cm?
9
A cuboid is 7 cm long, 3 cm wide and 2 cm high. What is its volume, in cm³?
10
A cube has sides of 0.3 m. What is its volume, in cm³?

Total 21 marks. Allow about 21 minutes.

Exercise 9X  SATs-style questions

Non-calculator
1
FChange 3.875 kilometres to metres.[1 mark]
2
FChange 2.75 hours into minutes.[1 mark]
3
FChange 2.4 metres to millimetres.[1 mark]
4
PSA parallelogram has a base of 14 cm, a perpendicular height of 6 cm and a slanted side of 9 cm. Find its area, in cm². Show your working.[2 marks]
5
PSA triangle has a base of 13 cm and a perpendicular height of 8 cm. Find its area, in cm². Show your working.[2 marks]
6
PSA race is 36 miles long. Use 1 mile = 1.6 km to find its length in kilometres. Show your working.[2 marks]
7
PSA cuboid is 8 cm long and 5 cm wide. Its volume is 200 cm³. How high is it, in cm? Show your working.[2 marks]
8
PSA rectangle has whole-number sides and an area of 48 cm². What is the smallest perimeter it can have, in cm? Show your working.[3 marks]
9
PSA tank is 1.2 m long, 0.5 m wide and 0.8 m high. What is its volume, in cm³? Show your working.[3 marks]
10
RA rectangle card is 8 cm by 6 cm. A corner measuring 3 cm by 2 cm is cut off.[4 marks]
(a)What is the area of the shape that is left, in cm²?
(b)Tia says that the perimeter of the shape left is the same as the perimeter of the whole card. Is Tia right? Explain your answer.
← Chapter 8: AlgebraChapter 10: Shapes, angles and circles →

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