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17.2 Circumference of a Circle 2

GCSE Maths Foundation · Edexcel · Chapter 17: Perimeter, Area and Volume 2

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: working backwards from the circumference, wheels and distance, rounding and error intervals. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Find the diameter or radius of a circle when you know its circumference
  • Solve problems about wheels, laps and turns using the circumference
  • Round answers to a given number of decimal places or significant figures
  • Write error intervals for rounded and truncated measurements

Key words

  • Inverse: The opposite operation. Dividing by π undoes multiplying by π.
  • Revolution: One complete turn of a wheel. In one turn a wheel rolls forward one circumference.
  • Error interval: The range of values a number could have been before it was rounded or truncated, for example 7.5 ≤ r < 8.5.
  • Truncate: Chop off digits after a certain place without rounding. 25.137 truncated to 1 dp is 25.1.
  • Significant figures: Digits counted from the first non-zero digit. 72.256 to 2 significant figures is 72.

From circumference back to diameter or radius

C = π × d, so to get back to d you do the inverse: divide by π. To get the radius, halve the diameter (or divide C by 2π).

Formulae: d = C ÷ π r = C ÷ (2 × π) = d ÷ 2

Worked example: a hoop

A hoop has circumference 50 cm. Find (a) its diameter and (b) its radius, to 1 decimal place.
(a) d = 50 ÷ π = 15.915… cm
(b) r = 15.915… ÷ 2 = 7.957… cm

Answer: (a) 15.9 cm (b) 8.0 cm

Tip: Keep the unrounded diameter in your calculator before halving. 15.9 ÷ 2 = 7.95 would round to a different answer.

Wheels, laps and turns

When a wheel turns once, it rolls forward a distance equal to its circumference.

Key fact: distance = number of turns × circumference number of turns = distance ÷ circumference

Worked example: distance travelled

A wheelchair wheel has diameter 66 cm. It makes 500 turns. How far does it travel, in metres?
Circumference = π × 66 = 207.345… cm
Distance = 500 × 207.345… = 103 672.5… cm = 1036.7 m

Answer: 1036.7 m (about 1.04 km)

Worked example: number of turns

A scooter wheel has diameter 0.7 m. How many complete turns does it make in 1 km?
Circumference = π × 0.7 = 2.199… m. Change 1 km to 1000 m so the units match.
1000 ÷ 2.199… = 454.7… The 455th turn is not finished.

Answer: 454 complete turns

Rounding answers

Circle answers are almost never whole numbers. Read the question to see how to round: decimal places (dp), significant figures (sf) or “to the nearest…”.

  • 72.256 63…; Instruction 1 decimal place; Answer 72.3
  • 72.256 63…; Instruction 3 significant figures; Answer 72.3
  • 72.256 63…; Instruction 2 significant figures; Answer 72
  • 72.256 63…; Instruction 1 significant figure; Answer 70

Worked example: a hula hoop

A hoop has radius 37 cm. Find its circumference to the nearest centimetre and in metres to 2 sf.
C = 2 × π × 37 = 232.477… cm

Answer: 232 cm, or 2.3 m to 2 sf

Error intervals

A measurement that has been rounded could have been a little bigger or smaller. The error interval shows every possible value. Go half a unit down and half a unit up.

  • r = 8 cm; Rounded to nearest cm (unit = 1); Error interval 7.5 ≤ r < 8.5
  • C = 47.1 cm; Rounded to 1 dp (unit = 0.1); Error interval 47.05 ≤ C < 47.15
  • d = 300 mm; Rounded to nearest 10 mm; Error interval 295 ≤ d < 305

The lower end uses ≤ because it rounds up to the value. The upper end uses < because exactly 8.5 would round up to 9.

Worked example: truncation

Dev's calculator shows a circumference of 25.132 74… He truncates it to 1 dp and writes 25.1.
Any number from 25.1 up to (but not including) 25.2 truncates to 25.1.

Answer: Error interval 25.1 ≤ x < 25.2

Common mistakes

  • Dividing by 2π when the question asks for the diameter. Dividing by 2π gives the radius.
  • Mixing units: a wheel measured in cm and a distance in km must be changed to the same unit first.
  • Rounding up the number of complete turns. 454.7 turns is 454 complete turns.
  • Writing ≤ at the top of an error interval. The upper bound is never included: 7.5 ≤ r < 8.5.
  • For truncation, going half a unit down. A truncated value is the lowest possible value.

Revision questions

Try these, then check your answers below each one.

Question 1. A circle has circumference 75 cm. Work out its diameter. Give your answer to 1 decimal place.

Answer: 23.9 cm

Working: d = 75 ÷ π = 23.873… = 23.9 cm.

Question 2. A circle has circumference 30 m. Work out its radius. Give your answer to 1 decimal place.

Answer: 4.8 m

Working: r = 30 ÷ (2π) = 4.774… = 4.8 m.

Question 3. The radius of a circle is 9 cm, correct to the nearest centimetre. Which is the error interval for the radius r? (Choose from: 8.5 ≤ r < 9.5, 8.5 < r ≤ 9.5, 8 ≤ r < 10, 8.9 ≤ r < 9.1.)

Answer: 8.5 ≤ r < 9.5

Working: Half of 1 cm is 0.5 cm, so 8.5 ≤ r < 9.5.

Related lessons

Continue in the full course

This lesson is section 17.2 (EGCMF1702) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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