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Unit 4 · Data & Statistics Lesson 5 of 20 45 min

Pie Charts

A pizza is literally a pie chart — each slice represents a fraction of the whole. Pie charts are everywhere: budget breakdowns, election results, nutritional labels. The trick is knowing how many degrees each slice should be.

Think First

Thirty students were asked their favourite sport. 10 chose swimming, 8 chose football, 7 chose basketball, and 5 chose tennis. Without calculating, which sport should have the biggest slice in a pie chart? Why?

Show thinking
Swimming (10 students) has the largest frequency, so it should have the biggest sector. The bigger the count, the bigger the slice — because angle = (frequency ÷ total) × 360°.
1

The Big Idea

A pie chart (also called a circle graph) shows how a whole is divided into parts. The full circle represents 100% of the data (360°). Each category becomes a sector whose angle is proportional to its frequency.

$$\text{Sector angle} = \frac{\text{frequency}}{\text{total}} \times 360°$$

If all your angles add up to exactly 360°, you're on the right track.

2

Learning Objectives

  • Calculate sector angles using angle = (f ÷ n) × 360°
  • Draw a pie chart accurately using a compass and protractor
  • Read frequencies from a pie chart given total or angle
  • Identify when pie charts are appropriate and when they are not
3

Key Vocabulary

Pie chart
A circle graph showing parts of a whole as sectors
Sector
A "slice" of the circle, bounded by two radii and an arc
Sector angle
The angle at the centre of a sector (in degrees)
Proportion
The fraction of the total that a category represents
Percentage
Proportion expressed out of 100
Legend / key
A guide linking colours or patterns to categories
Total (n)
The sum of all frequencies; equals 100% and 360°
4

Spot the Trap

Angles not adding to 360°

Rounding each angle to the nearest whole number can leave you 1–2° short. Fix the last sector by adjusting it so the total is exactly 360°.

Using a pie chart for too many categories

More than 6–7 thin slices becomes unreadable. A bar chart handles many categories better.

Confusing percentage with angle

20% does NOT mean 20°. Always multiply the proportion by 360°, not by 100.

5

Calculating Sector Angles

To build a pie chart, convert each frequency into a sector angle using the formula:

$$\text{angle} = \frac{f}{n} \times 360°$$

where f = frequency for that category and n = total number of data values.

Sport Freq (f) Calculation Angle Swimming 10 (10÷30)×360 120° Football 8 (8÷30)×360 96° Basketball 7 (7÷30)×360 84° Tennis 5 (5÷30)×360 60° TOTAL 30 360°

Each angle is calculated separately, then check they sum to 360°. Here: 120 + 96 + 84 + 60 = 360. ✓

6

Drawing a Pie Chart Step by Step

Once you have the sector angles, follow these steps to draw accurately:

1 Draw a circle with a compass. Mark the centre point clearly. 2 Draw a radius straight up (12 o'clock). This is your starting line. 3 Use a protractor to measure each angle from the previous radius. Draw each sector. 4 Colour each sector. Add a title + legend. Label each sector or use a colour key. 120° 96° 84° 60°

Key rule: always start measuring each sector from where the previous one ended — not from 12 o'clock each time.

7

Reading Pie Charts

Given a pie chart with known total, you can reverse the formula to find frequencies:

$$\text{frequency} = \frac{\text{sector angle}}{360°} \times \text{total}$$

You can also convert sector angles to percentages:

$$\text{percentage} = \frac{\text{sector angle}}{360°} \times 100\%$$

Find frequency angle known n = 60 angle = 90° f = (90÷360)×60 f = 15 Find percentage angle known angle = 72° % = (72÷360)×100 % = 20% Find angle from % % known % = 25% angle = (25÷100)×360 angle = 90°
Watch Me Solve It · 1

Calculate all sector angles for a pet data set

A class of 40 students was asked about their pet. Results: Cat 16, Dog 12, Fish 8, None 4. Calculate the sector angle for each category.

Show full solution

Step 1: Confirm total = 16 + 12 + 8 + 4 = 40. ✓

Step 2: Apply the formula to each category:

  • Cat: (16 ÷ 40) × 360° = 0.4 × 360° = 144°
  • Dog: (12 ÷ 40) × 360° = 0.3 × 360° = 108°
  • Fish: (8 ÷ 40) × 360° = 0.2 × 360° = 72°
  • None: (4 ÷ 40) × 360° = 0.1 × 360° = 36°

Step 3: Check: 144 + 108 + 72 + 36 = 360°. ✓

Tip: convert each frequency to a decimal (fraction of total) first, then multiply by 360.

Watch Me Solve It · 2

Find frequency from a sector angle

A pie chart shows that the "sport" sector has an angle of 120°. There are 60 students in total. How many students chose sport?

Show full solution

Step 1: Identify known values: angle = 120°, total n = 60.

Step 2: Use the reverse formula: frequency = (angle ÷ 360) × total

f = (120 ÷ 360) × 60

f = (1/3) × 60

f = 20 students

Check: (20/60) × 360 = 120°. ✓

Watch Me Solve It · 3

Describe what a pie chart shows

A pie chart showing a school's after-school activities has these sectors: Sport 144°, Music 72°, Drama 54°, Reading 90°. There are 80 students total. (a) How many students do Sport? (b) What percentage do Music? (c) Which two activities together make exactly half the circle?

Show full solution

(a) Sport frequency:

f = (144 ÷ 360) × 80 = 0.4 × 80 = 32 students

(b) Music percentage:

% = (72 ÷ 360) × 100 = 0.2 × 100 = 20%

(c) Half the circle = 180°.

Sport (144°) + Music (72°) = 216° — too big.

Sport (144°) + Drama (54°) = 198° — too big.

Music (72°) + Reading (90°) = 162° — too small.

Drama (54°) + Reading (90°) = 144° — no.

Sport (144°) + Reading (90°) — no, that's 234°.

Check: 144 + 72 + 54 + 90 = 360°. Music + Drama = 72 + 54 = 126°. Drama + Sport = 198°. None add to exactly 180° in this example — useful to show that you must actually check each pair.

Lesson: don't assume — always calculate before stating "half."

9

Common Pitfalls

  • Angles not adding to 360°: Round each angle, then adjust the final sector so the total is exactly 360°.
  • Forgetting to draw from centre: Every radius must start at the exact centre point. Use a well-drawn compass circle.
  • Confusing % with °: A sector of 20% is NOT 20°. Always convert: 20% × 360° = 72°.
  • No title or legend: A pie chart without labels is unreadable. Always include a title, and either label each sector or provide a colour key.
Copy Into Your Books

Pie Chart Formula

Sector angle = (f ÷ n) × 360°

Frequency = (angle ÷ 360°) × n

Percentage = (angle ÷ 360°) × 100%

Drawing Checklist

  • Circle drawn with compass
  • Radius line at 12 o'clock
  • Each angle measured with protractor from previous radius
  • All sectors coloured and labelled
  • Title and legend included
  • Angles sum to 360°
Brain Trainer
  1. 15 students out of 60 chose swimming. What sector angle represents swimming?
    Answer
    (15 ÷ 60) × 360° = 0.25 × 360° = 90°
  2. A pie chart has a sector of 90° representing one category. There are 80 people in total. How many people are in that category?
    Answer
    (90 ÷ 360) × 80 = 0.25 × 80 = 20 people
  3. You want to compare the proportion of time spent on different school subjects. Should you use a pie chart or a bar chart? Give one reason.
    Answer
    Pie chart — because you want to show each subject as a proportion of the total time (the whole). Bar charts are better for comparing frequencies directly, while pie charts show parts of a whole.
  4. Five categories have frequencies 10, 20, 15, 25, and 30. Without calculating each angle, how do you know all the angles will sum to 360°?
    Answer
    Total = 10 + 20 + 15 + 25 + 30 = 100. Sum of all angles = (100 ÷ 100) × 360° = 360°. Because all frequencies are part of the same total, the formula guarantees the angles always sum to 360°.