What Is a Circle Graph?

A circle graph, also called a pie chart, represents how a whole is divided into:

parts or categories.

The entire circle represents:

100% of the data.

It also represents:

360°.

Each sector, or slice, represents one category's share of the total.

This makes circle graphs especially useful when we want to answer:

How much of the whole belongs to each category?

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Parts of a Whole

Suppose 40 students are asked about their favourite school subject.

Each category represents part of the:

40 students.

For example:

Science = 12 out of 40 students

Mathematics = 10 out of 40 students

English = 8 out of 40 students

A circle graph converts these parts into:

sections of a circle.

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From Frequency to Percentage

To calculate the percentage represented by a category, use:

Percentage = (Category Frequency ÷ Total Frequency) × 100

For Science:

Percentage = (12 ÷ 40) × 100 = 30%

For Mathematics:

Percentage = (10 ÷ 40) × 100 = 25%

For English:

Percentage = (8 ÷ 40) × 100 = 20%

For Art:

Percentage = (6 ÷ 40) × 100 = 15%

For Music:

Percentage = (4 ÷ 40) × 100 = 10%

Check:

30% + 25% + 20% + 15% + 10% = 100%

Now the data can be represented as a circle graph.

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The Whole Is Always 100%

The most important idea in a circle graph is:

Whole = 100%

All sectors combined must represent:

100%.

For example:

25% + 35% + 20% + 20% = 100%

This means every observation has been accounted for.

If your percentages total:

137%

something is wrong.

If they total:

74%

some data may be missing, unless the chart deliberately represents only part of a larger total.

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A Circle Is 360°

A complete circle contains:

360°

Therefore:

100% = 360°

This relationship allows us to convert percentages into:

sector angles.

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Percentage and Angle

Some common percentages are useful to recognize.

These relationships make circle graphs easier to:

construct and interpret.

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Calculating Sector Angles

To convert a percentage into an angle:

Sector Angle = (Percentage ÷ 100) × 360°

For example, if a category represents 25%:

Sector Angle = (25 ÷ 100) × 360°

Sector Angle = 90°

So 25% of a circle is:

90°.

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From Frequency Directly to Angle

You do not have to calculate the percentage first.

You can use:

Sector Angle = (Category Frequency ÷ Total Frequency) × 360°

For example:

12 out of 40 students chose Science.

Sector Angle = (12 ÷ 40) × 360°

Sector Angle = 108°

Therefore, the Science sector should measure:

108°.

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Worked Example 1: Complete Circle Graph Calculations

Return to our favourite-subject data.

Check:

108° + 90° + 72° + 54° + 36° = 360°

The calculations are:

consistent.

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Three Equivalent Ways to Describe a Sector

A sector can be described using a:

fraction

percentage

or:

angle.

For Science:

Fraction = 12/40 = 3/10

Percentage = 30%

Angle = 108°

These all represent:

the same proportion of the whole.

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Constructing a Circle Graph

Suppose a survey produces:

Let's construct a circle graph from the data.

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Step 1: Find the Total

Add the frequencies:

15 + 10 + 8 + 7 = 40

Therefore:

Total = 40 students

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Step 2: Calculate Percentages

Bus

Percentage = (15 ÷ 40) × 100 = 37.5%

Car

Percentage = (10 ÷ 40) × 100 = 25%

Walk

Percentage = (8 ÷ 40) × 100 = 20%

Bicycle

Percentage = (7 ÷ 40) × 100 = 17.5%

Check:

37.5% + 25% + 20% + 17.5% = 100%

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Step 3: Calculate Angles

Bus

Angle = (37.5 ÷ 100) × 360° = 135°

Car

Angle = (25 ÷ 100) × 360° = 90°

Walk

Angle = (20 ÷ 100) × 360° = 72°

Bicycle

Angle = (17.5 ÷ 100) × 360° = 63°

Check:

135° + 90° + 72° + 63° = 360°

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Step 4: Draw the Circle

Use a compass or suitable digital tool to create a:

circle.

Mark its:

centre.

Draw one radius from the centre to the edge.

This provides your:

starting line.

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Step 5: Measure Each Sector

Use a protractor to measure:

135°

then:

90°

then:

72°

then:

63°.

The final sector should complete the:

360° circle.

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Step 6: Label the Graph

Each sector should be clearly identified.

You can:

• write the category inside the sector
• include the percentage
• use a legend or key
• use labels beside the graph

For example:

Bus — 37.5%

A clear graph should not require the reader to:

guess what the sectors mean.

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Step 7: Add a Title

Use a descriptive title such as:

How Students Travel to School

rather than simply:

Pie Chart.

The title should explain:

what the data represent.

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Interpreting Circle Graphs

Circle graphs allow us to compare categories using:

proportions.

A larger sector represents:

a larger share of the total.

A smaller sector represents:

a smaller share of the total.

For example:

Half of the circle = 50%

One quarter of the circle = 25%

Three quarters of the circle = 75%

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Worked Example 2: Reading a Circle Graph

Suppose a family budget is represented as:

Housing is the:

largest category.

Entertainment is the:

smallest category.

Housing represents four times the share of Entertainment because:

40% ÷ 10% = 4

Food represents twice the share of Entertainment because:

20% ÷ 10% = 2

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Finding an Actual Amount From a Percentage

Suppose the family's monthly budget is:

$4,000

Housing represents:

40%.

Calculate:

Housing = 40% of $4,000

Housing = 0.40 × $4,000

Housing = $1,600

Therefore:

$1,600 is allocated to housing.

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Finding Frequency From a Circle Graph

Suppose a circle graph shows that:

30%

of 200 students prefer basketball.

Calculate:

Number of Students = 0.30 × 200

Number of Students = 60

Therefore:

60 students prefer basketball.

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Finding a Percentage From an Angle

Suppose a sector measures:

72°.

Use:

Percentage = (Sector Angle ÷ 360°) × 100

Therefore:

Percentage = (72 ÷ 360) × 100

Percentage = 20%

The sector represents:

20% of the whole.

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Finding Frequency From an Angle

Suppose 120 students participated in a survey.

A sector representing football measures:

90°.

First calculate the fraction of the circle:

90 ÷ 360 = 1/4

Then calculate:

Number of Students = 1/4 × 120

Number of Students = 30

Therefore:

30 students chose football.

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Comparing Categories

Suppose a circle graph shows:

Science = 35%

Mathematics = 25%

English = 20%

Art = 15%

Music = 5%

How much greater is Science than Mathematics?

35% − 25% = 10 percentage points

The difference is:

10 percentage points.

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Percentage Points vs Percent Increase

Suppose Category A represents:

20%

and Category B represents:

30%.

The difference is:

30% − 20% = 10 percentage points

But the percentage increase from 20% to 30% is:

Percentage Increase = ((30 − 20) ÷ 20) × 100

Percentage Increase = 50%

Therefore:

10 percentage points

and:

50% increase

do not mean the same thing.

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Combining Categories

Suppose:

Food = 25%

Housing = 35%

Together:

25% + 35% = 60%

Therefore, Food and Housing represent:

60% of the total.

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Fractions and Circle Graphs

Circle graphs connect naturally to:

fractions.

Half of a circle:

1/2 = 50% = 180°

One quarter:

1/4 = 25% = 90°

Three quarters:

3/4 = 75% = 270°

One fifth:

1/5 = 20% = 72°

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Decimals, Fractions, and Percentages

These forms can all describe the same:

proportion.

For example:

1/4 = 0.25 = 25%

and:

3/5 = 0.60 = 60%

Circle graphs provide a visual way to connect:

fractions, decimals, percentages, and angles.

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When Are Circle Graphs Useful?

Circle graphs are especially useful when:

• categories form one meaningful whole
• the total can be treated as 100%
• there are relatively few categories
• the goal is to compare proportions
• differences between categories are reasonably visible

Examples include:

• household budgets
• survey responses
• market shares
• land use
• school enrolment by category
• spending categories
• energy sources

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When Are Circle Graphs Not Suitable?

Circle graphs are not appropriate for every:

dataset.

Suppose we record temperature:

These values do not represent parts of one:

whole.

A:

line graph

would be more appropriate.

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Circle Graphs Should Represent a Meaningful Whole

Consider the heights of four students:

150 cm

160 cm

170 cm

180 cm

Technically, we could add these values together and calculate percentages.

But what would the whole represent?

660 cm of student height?

That is not a meaningful part-to-whole quantity.

Therefore, a circle graph would be:

inappropriate.

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Too Many Categories

Suppose a dataset contains:

25 categories.

A circle graph would contain:

25 slices.

Many would be extremely small and difficult to:

distinguish.

In this situation, a:

bar graph

would usually communicate the information more clearly.

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Similar-Sized Categories

Suppose a circle graph contains:

21%, 20%, 20%, 19%, 20%

The sectors would look very:

similar.

Determining which category is slightly larger may be difficult.

A bar graph often makes small differences:

easier to compare.

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Circle Graph vs Bar Graph

The choice depends on:

what you want the reader to notice.

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Circle Graph vs Line Graph

A circle graph should not normally be used to show:

change over time.

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Worked Example 3: Choosing the Correct Graph

Situation A

A company wants to show how its annual spending is divided among salaries, rent, equipment, advertising, and utilities.

A:

circle graph

could be appropriate because the categories form:

one total budget.

Situation B

The company wants to show its revenue from January through December.

A:

line graph

would usually be more appropriate because the goal is to show:

change over time.

Situation C

The company wants to compare sales for 15 different products.

A:

bar graph

would probably be easier to read because there are:

many categories.

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Evaluating a Circle Graph

When evaluating a circle graph, ask:

Does the data form a whole?

If not, a circle graph may be inappropriate.

Do the percentages total approximately 100%?

Small rounding differences may occur, but large discrepancies indicate a:

problem.

Are the categories clearly labelled?

Every sector should be identifiable.

Are there too many sectors?

Too many slices reduce:

readability.

Are the sector sizes accurate?

The visual size should match the numerical:

proportion.

Is another graph type clearer?

A graph should be selected for communication, not simply because it can be:

constructed.

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Misleading Circle Graphs

Like other graphs, circle graphs can be designed in ways that distort:

visual comparisons.

Common problems include:

• incorrect sector sizes
• missing categories
• percentages that do not total 100%
• unclear labels
• unnecessary 3D effects
• too many slices
• visually emphasizing one category
• using a circle graph when the data do not form a whole

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The Problem With 3D Pie Charts

Three-dimensional effects can make sectors near the front appear:

larger.

Sectors near the back may appear:

smaller.

The numerical data have not changed, but perspective changes the:

visual impression.

For accurate comparisons, a simple two-dimensional circle graph is usually:

clearer.

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Worked Example 4: Spot the Problem

A pie chart shows:

Food = 35%

Housing = 40%

Transport = 20%

Entertainment = 15%

Calculate the total:

35% + 40% + 20% + 15% = 110%

This cannot represent one complete whole.

The chart contains:

an error.

The data or calculations should be checked.

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Worked Example 5: Rounding

Suppose three categories produce:

33.3%

33.3%

33.3%

Their total is:

99.9%.

This does not necessarily mean the graph is wrong.

The missing:

0.1%

may result from:

rounding.

Small rounding differences are normal.

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Worked Example 6: Favourite Sports

A survey of 80 students gives:

Football

Percentage = (28 ÷ 80) × 100 = 35%

Basketball

Percentage = (20 ÷ 80) × 100 = 25%

Swimming

Percentage = (16 ÷ 80) × 100 = 20%

Tennis

Percentage = (12 ÷ 80) × 100 = 15%

Other

Percentage = (4 ÷ 80) × 100 = 5%

Check:

35% + 25% + 20% + 15% + 5% = 100%

Now calculate the angles:

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Reverse Problems

Sometimes you know the sector but need to determine the:

original data.

Suppose a survey contains:

240 people.

A sector measures:

60°.

First:

Fraction of Circle = 60 ÷ 360 = 1/6

Then:

Number of People = 1/6 × 240 = 40

Therefore:

40 people are represented.

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Another Reverse Problem

A circle graph represents:

500 households.

A category represents:

18%.

Calculate:

Number of Households = 0.18 × 500

Number of Households = 90

Therefore:

90 households belong to that category.

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Comparing Two Circle Graphs

Be careful when comparing two different:

circle graphs.

Suppose School A has:

100 students

and 40% participate in sports.

40% of 100 = 40 students

School B has:

500 students

and 30% participate in sports.

30% of 500 = 150 students

Although School A has the larger:

percentage,

School B has the larger:

number of students.

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Percentages and Frequencies Are Different

A larger percentage does not always mean a larger:

frequency.

You must also know the:

total size of the group.

For example:

50% of 20 = 10

while:

25% of 200 = 50

The smaller percentage represents the:

larger number.

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Real-World Uses of Circle Graphs

Circle graphs appear in:

• business reports
• household budgets
• market research
• demographic summaries
• surveys
• environmental reports
• school statistics
• spending reports
• resource allocation

They are useful because people can quickly see:

how a total is divided.

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Circle Graphs in Science

Circle graphs can sometimes be useful in science.

For example, a scientist might display the proportion of organisms in a habitat:

Plants = 45%

Insects = 30%

Birds = 15%

Other organisms = 10%

But circle graphs are less useful for showing:

continuous experimental relationships.

For relationships such as:

temperature vs reaction rate

a line or scatter graph would normally be more appropriate.

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Constructing Circle Graphs Digitally

Spreadsheet software can automatically create:

pie charts.

Usually, you:

1. enter the categories
2. enter the values
3. select the data
4. choose a pie or circle graph
5. add labels
6. display percentages if useful
7. add a descriptive title

Digital tools make construction easier.

But the computer cannot decide whether a circle graph is:

the best representation.

That requires:

mathematical judgment.

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The PIE Check

Before creating or accepting a circle graph, use:

P — Parts

Do the categories represent parts of one meaningful whole?

I — Information

Are the labels, percentages, and categories clear?

E — Entire Whole

Do all categories together represent approximately 100%?

If the answer to one of these is no, reconsider whether the graph is:

appropriate.

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Check Your Understanding

1. What is another name for a circle graph?

2. What percentage does the entire circle represent?

3. How many degrees are in a complete circle?

4. Write the formula for calculating a category's percentage.

5. Write the formula for calculating a sector angle from frequency.

6. What angle represents 50%?

7. What angle represents 25%?

8. What percentage is represented by 72°?

9. A survey contains 60 students. If 15 choose basketball, what percentage chose basketball?

10. Calculate the sector angle for the basketball group in Question 9.

11. A category represents 35% of 200 people. How many people does it represent?

12. A sector measures 90°. What fraction and percentage of the circle does it represent?

13. Why should the percentages in a circle graph total approximately 100%?

14. Explain why a small rounding difference may be acceptable.

15. Why is a circle graph unsuitable for displaying temperature changes throughout a day?

16. Why might a bar graph be preferable when a dataset contains 20 categories?

17. Explain why 3D effects can make a circle graph misleading.

18. School A has 200 students and 40% play football. School B has 500 students and 25% play football. Which school has more football players? Show your calculations.

19. Explain the difference between a percentage and a percentage-point difference.

20. Give three questions you should ask when evaluating whether a circle graph is suitable for a dataset.

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Key Terms

• Circle graph: Circular graphical display showing how a whole is divided among categories.
• Pie chart: Another name for a circle graph.
• Whole: Complete quantity represented by the entire circle.
• Sector: Region or slice of a circle representing a category.
• Frequency: Number of observations belonging to a category.
• Fraction: Number representing part of a whole.
• Decimal: Base-ten representation of a numerical value.
• Percentage: Proportion expressed out of 100.
• Proportion: Comparative relationship between a part and a whole.
• Sector angle: Angle at the centre of a circle representing a category.
• Degree: Unit used to measure angles.
• Percentage point: Unit describing the arithmetic difference between two percentages.
• Part-to-whole relationship: Comparison of an individual category with the complete total.
• Rounding: Replacing a value with a nearby value containing fewer digits.
• Graphical representation: Visual method of communicating data.

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Key Takeaways

• A circle graph is also called a pie chart.
• Circle graphs show how a meaningful whole is divided into parts.
• The complete circle represents 100%.
• A complete circle contains 360°.
• Each sector represents one category's proportion of the total.
• Calculate percentage using: Percentage = (Category Frequency ÷ Total Frequency) × 100.
• Calculate sector angle using: Sector Angle = (Category Frequency ÷ Total Frequency) × 360°.
• A sector's fraction, decimal, percentage, and angle all describe the same proportion.
• 50% = 180°.
• 25% = 90°.
• 20% = 72°.
• 10% = 36°.
• Percentages should total approximately 100%.
• Sector angles should total 360°.
• Small differences from 100% may result from rounding.
• Circle graphs are most effective when the categories form one meaningful whole.
• Circle graphs work best with a relatively small number of categories.
• Too many sectors make a circle graph difficult to read.
• Bar graphs are usually better when precise comparisons between many categories are required.
• Line graphs are usually better for showing change over time.
• Percentage data can be converted back into frequencies when the total is known.
• A larger percentage does not necessarily represent a larger number when two datasets have different totals.
• 3D effects and distorted sectors can make circle graphs misleading.
• An effective circle graph should be accurate, clearly labelled, easy to interpret, and appropriate for the dataset.