What is a pie chart?
A pie chart shows data as proportional sectors ("slices") of a circle. Each sector's angle represents its share of the total.
Key facts:
- A full circle is 360°
- Angle for a category = (frequency ÷ total frequency) × 360°
- All the sector angles in a pie chart must add up to exactly 360° — use this to check your work
- A bigger sector (larger angle) always represents a larger proportion of the data — but not necessarily a larger actual number, if you're comparing two different pie charts with different totals
Calculating angles from a table
Worked Example 1
The table shows the favourite sport of 90 students. Calculate the angle for each sector.
| Sport | Football | Rugby | Tennis | Swimming |
|---|---|---|---|---|
| Frequency | 36 | 18 | 9 | 27 |
| Angle | 144° | 72° | 36° | 108° |
1
Football: (36 ÷ 90) × 360° = 144°
2
Rugby: (18 ÷ 90) × 360° = 72°. Tennis: (9 ÷ 90) × 360° = 36°. Swimming: (27 ÷ 90) × 360° = 108°
3
Check: 144 + 72 + 36 + 108 = 360° ✓
Answer144°, 72°, 36°, 108°
Each sector's angle is proportional to its frequency. The four angles add up to 360°, confirming the pie chart accounts for all 90 students.
Finding a frequency from an angle
Worked Example 2
In a pie chart representing 60 people's favourite drink, the "tea" sector has an angle of 90°. Find the number of people who chose tea.
1
Find the fraction of the circle: 90° ÷ 360° = 1/4
2
Multiply by the total: 1/4 × 60 = 15
Answer15 people
Finding the total from one sector
Worked Example 3
In a pie chart, the "walking" sector has an angle of 60° and represents 15 people. Find the total number of people surveyed.
1
15 people correspond to 60° out of 360°
2
Scale up to the full circle: total = 15 × (360° ÷ 60°) = 15 × 6
3
= 90 people
Answer90 people
Comparing two pie charts
Worked Example 4 — A common trap
Two pie charts show favourite subject for Year 10 (200 students) and Year 11 (150 students). In both pie charts, the "Maths" sector has the same angle, 90°. Does the same number of students in each year prefer Maths?
1
Year 10: (90° ÷ 360°) × 200 = 50 students
2
Year 11: (90° ÷ 360°) × 150 = 37.5 students
3
Even though the angles — and therefore the slice sizes — look identical, the actual numbers of students are different, because the two pie charts represent different total group sizes
AnswerNo — 50 students in Year 10, but only 37.5 (≈38) in Year 11
Practice questions
Work through each question before checking the answers.
Foundation (Grade 3–5)
Q1A pie chart represents 120 people's favourite fruit. The apple sector has an angle of 90°. Find the number of people who chose apple.Foundation
Q2In a survey of 200 people, 40 chose "walking" as their favourite exercise. Find the angle for the walking sector in a pie chart.Foundation
Q3A pie chart has four sectors with angles 120°, 90°, 60° and x°. Find the value of x.Foundation
Q4A pie chart shows the results of a survey of 80 people. A sector has an angle of 45°. How many people does it represent?Foundation
Q5True or false: in a pie chart, a larger sector always represents a smaller proportion of the data. Explain your answer.Foundation
Higher (Grade 5–7)
Q6
The table shows the favourite pet of 150 people. Find the angle for the "Cat" sector.
Higher| Pet | Dog | Cat | Fish | Other |
|---|---|---|---|---|
| Frequency | 60 | 45 | 25 | 20 |
Q7Using the data from Q6, find the angle for the "Fish" sector.Higher
Q8Using the data from Q6, find the angle for the "Other" sector, and check that all four angles sum to 360°.Higher
Q9In a pie chart representing 90 students, the "Bus" sector has an angle of 100°. Find the number of students who travel by bus.Higher
Q10A pie chart has sectors for red (angle 150°), blue (angle 130°), and green. Find the angle for green.Higher
Higher — Hard (Grade 8–9)
Q11Two pie charts show favourite subject for Year 10 (200 students) and Year 11 (150 students). In both pie charts, "Maths" has an angle of 90°. Determine whether the same number of students in each year chose Maths, showing your working.Grade 8–9
Q12A pie chart represents a survey of n people. The "yes" sector has an angle of 150° and represents 75 people. Find n.Grade 8–9
Q13In a pie chart, the angles for three categories A, B and C are in the ratio 2 : 3 : 4. Find the angle for category B.Grade 8–9
Q14A pie chart shows the results of a survey of x people. A sector with angle 72° represents 18 people. Find x.Grade 8–9
Q15Explain why it can be misleading to compare two pie charts with different total sample sizes using only the size of their sectors, using a specific example to support your answer.Grade 8–9
Answers
Foundation (Q1–Q5)
Q130(90/360 × 120)
Q272°(40/200 × 360)
Q3x = 90(360 − 120 − 90 − 60)
Q410(45/360 × 80)
Q5False(a larger sector always represents a larger proportion of the data)
Higher (Q6–Q10)
Q6108°(45/150 × 360)
Q760°(25/150 × 360)
Q848°(20/150 × 360; check: 144 + 108 + 60 + 48 = 360 ✓)
Q925 students(100/360 × 90)
Q1080°(360 − 150 − 130)
Higher — Hard (Q11–Q15)
Q11No(Year 10: 90/360 × 200 = 50; Year 11: 90/360 × 150 = 37.5 — different totals mean equal angles don't give equal frequencies)
Q12n = 180(75 × 360/150 = 75 × 2.4)
Q13120°(9 parts = 360°, so 1 part = 40°; B = 3 × 40°)
Q14x = 90(18 × 360/72 = 18 × 5)
Q15Two pie charts can show sectors of the same angle (and therefore the same visual size) while representing very different actual numbers, if their total sample sizes differ — for example, a 90° "Maths" sector represents 50 students out of 200, but only 37.5 out of 150. Comparing sector size alone, without checking the totals, can give a misleading impression.
Common mistakes
Common Mistake 1
Forgetting to divide by the total before multiplying by 360°
The angle formula is (frequency ÷ total) × 360°, not frequency × 360°. Skipping the division gives a number far larger than 360°, which is impossible for a single sector.
Common Mistake 2
Not checking that the angles sum to 360°
After calculating every sector's angle, always add them up. If they don't total exactly 360° (allowing for small rounding), you've made an arithmetic error somewhere and should go back and check.
Common Mistake 3
Assuming equal angles mean equal frequencies across different charts
A sector with the same angle in two different pie charts only represents the same number of items if both pie charts have the same total. Always check the totals before comparing raw frequencies between charts.
Common Mistake 4
Mixing up the "find frequency" and "find total" methods
Finding a frequency from an angle uses (angle ÷ 360°) × total. Finding the total from one known sector uses frequency × (360° ÷ angle). Check which value the question gives you and which one it's asking for before setting up your calculation.
Exam tips
💡 Exam Tip 1
Use the 360° check as your first move
Before doing anything else, write down "angles must sum to 360°" — it immediately tells you how to find a missing angle by subtraction, and gives you a way to check your other answers are correct.
💡 Exam Tip 2
Use a protractor accurately when drawing by hand
When actually drawing a pie chart, measure each angle from the same starting line (usually straight up, like a clock's 12), and measure cumulatively round the circle rather than restarting from zero each time — this avoids small errors adding up.
💡 Exam Tip 3
Watch for questions that give the angle, not the frequency
Read carefully whether you're given a frequency and asked for an angle, or given an angle and asked for a frequency (or total) — the two calculations use the same ratio but in opposite directions.
💡 Exam Tip 4
Name the totals explicitly when comparing two pie charts
When a question asks you to compare two pie charts, always state both total sample sizes in your answer — this is usually where the key insight (and the marks) lie, especially when angles look deceptively similar.
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