Circumference of a Circle, and Miles and Kilometres

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

1. Key words

2. What is π?

Take any circle: a coin, a cup, a plate, a bicycle wheel. Measure its circumference (wrap a string round it) and its diameter. Then work out circumference ÷ diameter.

For every circle, big or small, C ÷ d gives the same number, just over 3. (The small differences in the table come from measuring errors.) This number is called π (the Greek letter pi). Its decimal digits never end and never repeat: π = 3.14159… We usually use π ≈ 3.14 or the π key on a calculator.

3. Finding the circumference, step by step

1. Read the question: is the length a radius or a diameter? If it is a radius, double it (or use C = 2πr).
2. Write the formula: C = πd.
3. Substitute: for example C = π × 12.
4. Calculate. With a calculator use the π key. Without a calculator use the value the question gives (often 3.14).
5. Round as the question asks (for example to 1 decimal place) and write the unit (cm, m…). Circumference is a length, so the unit is NOT squared.
6. Check: the answer should be a bit more than 3 × the diameter.

Perimeter of a semicircle = half the circumference + the diameter (the straight edge). Many students forget the straight edge.

4. Miles and kilometres

Distances between towns are measured in kilometres in most countries (Egypt uses km), but in some countries, such as the UK and the USA, road signs and speeds use miles. A mile is longer than a kilometre.

Check your answer makes sense: the number of kilometres is always bigger than the number of miles for the same distance (because a kilometre is the shorter unit, you need more of them).

Because miles and kilometres are in direct proportion, a conversion graph is a straight line through (0, 0). To use it, go up from the value on one axis to the line, then across to the other axis.

5. Worked examples

Worked example 1 [2 marks] (calculator) A circular clock face has diameter 12 cm. Calculate its circumference. Give your answer correct to 1 decimal place.

1. C = πd = π × 12
2. = 37.699… (calculator)
3. Answer: 37.7 cm (1 mark for π × 12 seen, 1 mark for 37.7).
4. Check: 3 × 12 = 36, and 37.7 is a bit more. ✓

Worked example 2 [2 marks] (calculator) A circular flower bed has radius 4.5 m. Calculate its circumference to 1 decimal place.

1. Radius given, so d = 2 × 4.5 = 9 m (or use C = 2πr).
2. C = π × 9 = 28.274…
3. Answer: 28.3 m.

Common mistake: π × 4.5 = 14.1 m. That uses the radius as if it were the diameter, giving only half the circumference.

Worked example 3 [1 mark] (no calculator) Use π = 3.14 to work out the circumference of a circle with diameter 10 cm.

C = 3.14 × 10 = 31.4 cm. (Multiplying by 10 moves each digit one place to the left.)

Worked example 4 [2 marks] (calculator) The circumference of a round tray is 50 cm. Work out its diameter, correct to 1 decimal place.

1. C = πd, so d = C ÷ π.
2. d = 50 ÷ π = 15.915…
3. Answer: 15.9 cm. (If the question asks for the radius: 15.915… ÷ 2 = 7.96 → 8.0 cm.)

Worked example 5 [3 marks] (calculator) The diagram shows a semicircle with diameter 8 cm. Calculate its perimeter to 1 decimal place.

1. Full circumference = π × 8 = 25.132…
2. Curved part = half of that = 12.566…
3. Perimeter = curved part + straight diameter = 12.566… + 8 = 20.566…
4. Answer: 20.6 cm. (1 mark for π × 8 ÷ 2, 1 mark for adding 8, 1 mark for 20.6.)

Worked example 6 [3 marks] (calculator) A bicycle wheel has diameter 70 cm. How far does the bicycle travel when the wheel turns 100 times? Give your answer in metres, to the nearest metre.

1. One turn moves the bicycle one circumference: π × 70 = 219.91… cm.
2. 100 turns: 219.91… × 100 = 21 991 cm.
3. Convert: 21 991 ÷ 100 = 219.9 m.
4. Answer: 220 m.

Worked example 7 [1 mark] A road sign in England says 'London 45 miles'. About how many kilometres is this? 45 × 1.6 = 45 + (45 × 0.6) = 45 + 27 = 72 km.

Worked example 8 [2 marks] A journey is 120 km. Use 1 km ≈ 5/8 mile to change this to miles. 120 ÷ 8 = 15, then 15 × 5 = 75 miles. Check: 75 is smaller than 120 ✓ (miles are fewer).

Worked example 9 [2 marks] Route A is 50 miles long. Route B is 85 km long. Which route is shorter, and by how many kilometres? Change to the same unit: 50 miles ≈ 50 × 1.6 = 80 km. 80 km < 85 km, so Route A is shorter, by about 5 km.

Common mistake: comparing 50 with 85 directly. You must change both distances to the same unit first.

Practice

All pages in this course

Start here

• Stage 8 Maths: The Tests and How to Prepare

Number

• Order of Operations and Calculating with Integers
• Factors, Multiples, Primes, HCF and LCM
• Types of Number and Index Laws
• Squares, Cubes and Roots (Positive and Negative)
• Multiplying and Dividing by 0.1 and 0.01, and Significant Figures
• Fractions: Recurring Decimals, Mixed Numbers and Order of Operations
• Percentage Increase, Decrease and Absolute Change
• Ordering and Calculating with Decimals
• Ratio and Proportion

Algebra

• Algebraic Expressions: Simplifying, Expanding and Factorising
• Writing Expressions and Formulae, and Changing the Subject
• Equations and Inequalities
• Sequences and the nth Term
• Functions and Straight-line Graphs (y = mx + c)
• Reading and Interpreting Real-life Graphs

Geometry and measure

• Quadrilaterals and Regular Polygons
• Circumference of a Circle, and Miles and Kilometres
• Area of Parallelograms and Trapezia
• 3D Shapes: Euler's Formula and Plans and Elevations
• Volume of Prisms and Surface Area
• Angles in Triangles and on Parallel Lines
• Constructions with Ruler and Compasses
• Bearings and Midpoints
• Translation, Reflection, Rotation and Enlargement

Statistics and probability

• Data Collection and Sampling
• Representing Data
• Averages, Range and Comparing Distributions
• Probability: Complementary and Combined Events

Practice tests

• Stage 8 Maths — Practice Test 1
• Stage 8 Maths — Practice Test 2
• Stage 8 Maths — Practice Test 3