Proportional Reasoning-2 · Lesson 5 of 7
Pie Charts — Proportions of a Whole
“Connect ratios, fractions, percentages and angles to construct and interpret pie charts as proportional pictures of a whole.”
- Explain how a pie chart represents parts of a whole.
- Convert data values into central angles out of 360°.
- Simplify a multi-term ratio before calculating slice angles when useful.
- Construct a pie chart with a ruler and protractor.
- Interpret fractions and counts from given pie-chart angles.
Why a Circle Can Represent a Whole
A pie chart uses one full circle to represent an entire data set. Each slice represents one category, and the size of its central angle is proportional to that category's share of the whole. Larger data values therefore produce larger slices.
A full circle measures 360°. So constructing a pie chart is really a ratio-division problem: we divide 360° among the categories in the same ratio as their data values.
From a Data Table to Slice Angles
Consider five groups with counts 12, 10, 8, 6 and 4. Their total is 40. We can calculate each angle directly from its fraction of 40, or simplify the ratio 12 : 10 : 8 : 6 : 4 to 6 : 5 : 4 : 3 : 2 and then divide 360° according to those 20 ratio-parts.
| Category | Count | Simplified parts | Angle |
|---|---|---|---|
| A | 12 | 6 | 108° |
| B | 10 | 5 | 90° |
| C | 8 | 4 | 72° |
| D | 6 | 3 | 54° |
| E | 4 | 2 | 36° |
Problem
Data values are 12, 10, 8, 6 and 4. Find the central angle for each category.
- 1.Add the values: 12 + 10 + 8 + 6 + 4 = 40.
- 2.For A: 12/40 × 360° = 108°.
- 3.For B: 10/40 × 360° = 90°.
- 4.For C: 8/40 × 360° = 72°.
- 5.For D: 6/40 × 360° = 54°.
- 6.For E: 4/40 × 360° = 36°.
- 7.Check: 108° + 90° + 72° + 54° + 36° = 360°.
Constructing the Pie Chart
After the angles have been calculated, the chart can be drawn one slice at a time. Accuracy depends on measuring from the same centre and using each new radius as the starting side for the next angle.
- Draw a circle and mark its centre.
- Draw one starting radius from the centre to the circle.
- Place the protractor at the centre and measure the first calculated angle.
- Draw the next radius to make the first slice.
- From that radius, measure the next angle and continue around the circle.
- Check that the final slice closes the circle and that all angles total 360°.
- Label each slice clearly; colour may be added to help distinguish categories.
Problem
Among 360 people, 90 prefer summer and 120 prefer rainy weather. The rest prefer winter. Find the angles for a pie chart.
- 1.Winter count = 360 − 90 − 120 = 150.
- 2.Because the total count is 360, each person corresponds to 1° in this particular data set.
- 3.Summer angle = 90°.
- 4.Rainy angle = 120°.
- 5.Winter angle = 150°.
- 6.Check: 90° + 120° + 150° = 360°.
Problem
A viewing survey gives Entertainment 50%, Sports 25%, News 15% and Information 10%. Find the pie-chart angles.
- 1.Entertainment: 50% of 360° = 180°.
- 2.Sports: 25% of 360° = 90°.
- 3.News: 15% of 360° = 54°.
- 4.Information: 10% of 360° = 36°.
- 5.The percentages total 100% and the angles total 360°.
Reading Information Back from a Pie Chart
A pie chart can also be read in reverse. A slice angle tells us the fraction of the whole, and if one category's count is known we can often recover the total number of observations.
Problem
A transport pie chart has Bus 120°, Walk 90°, Cycle 60°, Two-wheeler 60° and Car 30°. If 18 children travel by car, find the total number of children and the number using two-wheelers.
- 1.Car represents 30/360 = 1/12 of the whole.
- 2.If 1/12 corresponds to 18 children, the total is 18 × 12 = 216 children.
- 3.Two-wheeler represents 60/360 = 1/6 of the whole.
- 4.Number using two-wheelers = 216 × 1/6 = 36.
- 5.Bus has the largest angle, so it is the most common mode in this chart.
Do not use the raw data value as an angle unless the total happens to be 360. In general, convert each category's fraction of the total into the same fraction of 360°.
Quiz
A category contains 8 out of 40 observations. What angle should its pie-chart slice have?
A pie-chart slice measures 90°. What fraction of the whole does it represent?
Which set of angles can form a complete pie chart?
A category is 15% of the whole. What is its central angle?
In a pie chart, two categories both have 60° slices. What can we conclude?
Practice Problems
- A group of 360 people choose among rainy, winter and summer. Summer is chosen by 90 people and rainy by 120. Find the winter count and draw a pie chart.
- Convert these percentages into pie-chart angles: Entertainment 50%, Sports 25%, News 15%, Information 10%.
- Survey your class for one favourite subject per student. Make a frequency table, calculate each slice angle and construct a pie chart.
- A pie chart has slices of 144°, 108°, 72° and 36°. Write each slice as a fraction and a percentage of the whole.
- A transport chart has angles Bus 120°, Walk 90°, Cycle 60°, Two-wheeler 60° and Car 30°. Which mode is most common? What fraction travel by car? Which two modes have equal counts?
- Eight circles represent sleep as a fraction of a 24-hour day. If the sleep times are 15, 2.5, 20, 8, 3.5, 13, 10.5 and 18 hours, calculate the corresponding central angles that would represent sleep in each 24-hour circle.
Key Takeaways
A pie chart represents a whole circle of 360°. Each slice angle is proportional to that category's share of the total. Slice angle = category value ÷ total value × 360°. Ratios can be simplified before the angles are calculated. All slice angles must add to 360°. A slice angle can be converted back into a fraction by dividing it by 360°.