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Maths · Area and volume

Units and accuracy

Converting units of area, volume and capacity - where 1 m really is 100 cm but 1 m² is 10 000 cm² - writing error intervals for rounded measurements, and at Higher, using bounds in calculations.

  • 5 key terms
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Teacher resources

The teacher copies: slides with the questions built in, the answers, and anything else attached to this lesson for whoever is teaching it.

Student handouts

The same files the students see, to print or hand out.

Last Lesson and Before

Answer each one, then check.

  1. 1

    How many cm are in 1 m?

    Show answerHide answer

    100

  2. 2

    How many mm are in 1 cm?

    Show answerHide answer

    10

  3. 3

    Round 8.347 to 1 decimal place.

    Show answerHide answer

    8.3

  4. 4

    Last lesson: area of a 3 m by 2 m rectangle?

    Show answerHide answer

    6 m²

Learning Objectives

  1. 1Convert between units of area.
  2. 2Convert between units of volume and capacity.
  3. 3Write the error interval for a rounded or truncated number.
  4. 4(Higher) Find upper and lower bounds of a calculation.

Conversion Factors

  • mm and cm

    Length: 1 cm = 10 mm. Area: 1 cm² = 100 mm². Volume or capacity: 1 cm³ = 1000 mm³

  • cm and m

    Length: 1 m = 100 cm. Area: 1 m² = 10 000 cm². Volume or capacity: 1 m³ = 1 000 000 cm³

  • m and km

    Length: 1 km = 1000 m. Area: 1 km² = 1 000 000 m². Volume or capacity: Rarely needed

  • Capacity

    Length: -. Area: -. Volume or capacity: 1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1000 litres

Converting Area

Convert 3.5 m² into cm².

Show the solutionHide the solution
  1. 1 1 m² = 10 000 cm² Multiply by 10 000 (a smaller unit means more of them)
  2. 2 Calculate \(3.5 \times 10\,000 = 35\,000\)

Answer35 000 cm²

Converting Volume to Capacity

A tank holds 2 400 000 cm³ of water. Write this in m³ and in litres.

Show the solutionHide the solution
  1. 1 1 m³ = 1 000 000 cm³ \(2\,400\,000 \div 1\,000\,000 = 2.4\) m³
  2. 2 1 litre = 1000 cm³ \(2\,400\,000 \div 1000 = 2400\) litres
  3. 3 Check: 1 m³ = 1000 litres \(2.4 \times 1000 = 2400\)

Answer2.4 m³, which is 2400 litres

Error Intervals

If a length is 8.3 cm to 1 decimal place, it could be anything that rounds to 8.3.

  • The bounds

    Half a unit either side: the lower bound is 8.25 and the upper bound is 8.35.

  • Error interval

    \(8.25 \le l < 8.35\). The upper bound itself would round up to 8.4, so use \(<\).

  • To the nearest 10

    70 to the nearest 10: \(65 \le x < 75\).

  • Truncation

    4.7 truncated to 1 d.p. (digits chopped off): \(4.7 \le x < 4.8\).

An Error Interval

The mass of a parcel is 2.6 kg, correct to 1 decimal place. Write down the error interval for the mass \(m\).

Show the solutionHide the solution
  1. 1 The degree of accuracy 0.1 kg
  2. 2 Half of it 0.05 kg
  3. 3 Lower and upper bounds \(2.6 - 0.05 = 2.55\) and \(2.6 + 0.05 = 2.65\)

Answer\(2.55 \le m < 2.65\)

Which Bounds to Use

  • \(a + b\)

    For the upper bound: upper + upper. For the lower bound: lower + lower

  • \(a - b\)

    For the upper bound: upper − lower. For the lower bound: lower − upper

  • \(a \times b\)

    For the upper bound: upper × upper. For the lower bound: lower × lower

  • \(a \div b\)

    For the upper bound: upper ÷ lower. For the lower bound: lower ÷ upper

The Upper Bound of a Speed

A runner covers 100 m, measured to the nearest metre, in 12.5 s, measured to 1 decimal place. Work out the upper bound of the runner's average speed.

Show the solutionHide the solution
  1. 1 Bounds of the distance 99.5 m and 100.5 m
  2. 2 Bounds of the time 12.45 s and 12.55 s
  3. 3 Biggest speed: biggest distance ÷ smallest time \(\dfrac{100.5}{12.45} = 8.0722\ldots\)

Answer8.07 m/s (to 3 s.f.)

Fill the Room

Measure (or estimate) your classroom's length, width and height in metres. Work out its volume in m³ and in litres. Then work out its floor area in cm². Which of your answers are sensible to give exactly, and which should be rounded?

1. Measure in metres.

2. Convert with the right factor.

3. Decide how accurate your answer really is.

A good answer shows: For a 9 m by 7 m by 3 m room: volume 189 m³ = 189 000 litres; floor area 63 m² = 630 000 cm². With measurements to the nearest metre, the answers are only good to 2 significant figures at most.

Can I...?

  1. 1Convert between units of area.
  2. 2Convert between units of volume.
  3. 3Convert between volume and capacity.
  4. 4Write an error interval for a rounded number.
  5. 5Write an error interval for a truncated number.
  6. 6(Higher) Find the bounds of a calculation.

Summary & Exam Focus

  • Area: square the length factor. Volume: cube it.
  • 1 cm³ = 1 ml; 1000 cm³ = 1 litre; 1 m³ = 1000 litres.
  • Error interval: half a unit either side, \(\le\) then \(<\).
  • (Higher) Maximise or minimise each part of the calculation.

Exam focus

The length of a pencil is 14 cm, correct to the nearest centimetre. Write down the error interval for the length \(l\). (2 marks) (2 marks)

In an error interval the lower bound gets \(\le\) and the upper bound gets \(<\). Writing \(\le\) at both ends loses a mark.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Capacity
How much a container holds, in ml or litres.
Lower bound
The smallest value that rounds to the given number.
Upper bound
The value where rounding switches up to the next number; the interval stops just below it.
Error interval
The range of possible values, e.g. \(8.25 \le l < 8.35\).
Truncate
Cut off digits without rounding.

Questions and answers

7 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Non-calculator 1 mark Easier

Change 3.5 m² into cm².

Mark scheme — 1 mark available

  • 35 000 — B1

Model answer

\(3.5 \times 10\,000 = 35\,000\) cm²

2. Exam question Non-calculator 2 marks Easier

A fish tank is a cuboid 80 cm long, 50 cm wide and 60 cm high. How many litres of water does it hold when full?

Mark scheme — 2 marks available

  • \(80 \times 50 \times 60\) — M1
  • 240 litres — A1

Model answer

\(80 \times 50 \times 60 = 240\,000\) cm³. \(240\,000 \div 1000 = 240\) litres.

3. Exam question Non-calculator 2 marks Easier

The length of a pencil, \(l\) cm, is 14 cm correct to the nearest centimetre. Write down the error interval for \(l\).

Mark scheme — 2 marks available

  • 13.5 and 14.5 — B1
  • \(13.5 \le l < 14.5\) — B1

Model answer

\(13.5 \le l < 14.5\)

4. Exam question Calculator · Higher 3 marks Easier

A runner covers a distance of 100 m, measured to the nearest metre, in a time of 12.5 s, measured to 1 decimal place. Work out the upper bound of the runner's average speed. Give your answer to 3 significant figures.

Mark scheme — 3 marks available

  • 100.5 or 12.45 seen — B1
  • \(\dfrac{100.5}{12.45}\) — M1
  • 8.07 — A1

Model answer

Upper bound \(= \dfrac{100.5}{12.45} = 8.0722\ldots = 8.07\) m/s

5. Multiple choice 1 mark Easier

How many cm² are in 1 m²?

  1. A 100
  2. B 1000
  3. C 1 000 000
  4. D 10 000 Correct

Why: 1 m² is 100 cm by 100 cm, which is \(100 \times 100 = 10\,000\) cm².

6. Multiple choice 1 mark Core

A length is 30 cm to the nearest 10 cm. What is the error interval?

  1. A \(29.5 \le x < 30.5\)
  2. B \(25 \le x < 35\) Correct
  3. C \(20 \le x < 40\)
  4. D \(25 < x \le 35\)

Why: Half of 10 is 5 either side: \(25 \le x < 35\).

7. Multiple choice 1 mark Stretch

(Higher) \(a = 6\) and \(b = 2\), both to the nearest whole number. What is the upper bound of \(a - b\)?

  1. A 4
  2. B 4.5
  3. C 5 Correct
  4. D 3

Why: Upper bound of \(a\) minus lower bound of \(b\): \(6.5 - 1.5 = 5\).