KS3 · Maths

Constructing and interpreting pie charts

Two pie charts, each with a 90° sector. On one it's a single student. On the other it's fifty. Same angle, so what is a pie chart really showing?

Maths · Pie charts

One multiplier, one circle

Frequency: 10/40. Angle 90 degrees. Frequency 10 students

Fraction10/40Angle90 degreesFrequency10 students

The bar is cut into 40 equal slices, one for each of 40 students, and the whole bar stands for the whole 360° circle. Drag the frequency and watch its angle keep pace: every student is worth the same 9°, because 360 ÷ 40 = 9.

Watch out: The 360° belongs to the total of THIS chart. A chart with a different total has a different multiplier.

Worked example: from a frequency table to sector angles

Problem

30 students say how they get to school: Walk 12, Bus 9, Cycle 6, Car 3. Work out the angle for each sector. Then try a trickier table: 33 students pick a weekend activity — Sport 12, Gaming 8, Music 7, Reading 6.

Maths · Pie charts

Where does this working go wrong?

A student draws a pie chart for 24 students' favourite sport: Football 9, Netball 6, Swimming 5, Tennis 4. Here is their working. Which line loses them the mark?

A student's working — which line goes wrong?

Total frequency: 9 + 6 + 5 + 4 = 24
Multiplier: 360 ÷ 24 = 15
Angles: Football 9°, Netball 6°, Swimming 5°, Tennis 4°
Check: 9 + 6 + 5 + 4 = 24°
So the four sectors use up just 24° of the circle.

Maths · Pie charts

Draw it in the right order

the order you do the construction steps in

1 · Do this first6 · Do this last

Label every sector
Measure the first angle, mark it with a cross and draw a radius through the cross
Check the final leftover sector by measuring its angle
Draw a circle, mark its centre and draw a radius from the centre to the edge
For each next sector, line the protractor's zero up with the line you drew most recently, then measure
Place the protractor's centre on the circle's centre, with its zero on that radius

Maths · Pie charts

Now run it backwards

A pie chart shows a year group's favourite lessons: Maths 120°, Science 90°, PE 60°, Art 30° and Music, whose angle isn't written on the chart. We know 5 students chose Art. Find how many students chose each lesson.

Find a row where you know both the frequency and the angle: Art has 5 students and 30°.If no angles are written on the chart, measure each one with a protractor, moving it so its zero lines up with a different radius each time. If you only know the total, the multiplier is 360 ÷ total.
missing step

Which line is step 2?

Multiplier = angle ÷ frequency = 30 ÷ 5 = 6Multiplier = 360 ÷ 5 = 72Multiplier = 5 ÷ 30 = 1/6

Maths · Pie charts

Same angle, same number?

Chart A is about the 4 students in a small club. Chart B is about the 200 students in a year group. In both charts, the 'walk to school' sector is 90°.

What do the two 90° sectors tell you? Pick the idea closest to what you think right now.

They're the same angle, so the same number of students walk in both charts.A bigger sector always means more students, even when I'm comparing two different charts.The angles tell me how many students walk, so I can read the frequencies straight off without anything else.Both sectors are the same proportion of their own chart. How many students that is depends on each chart's total, so I need a total or one frequency for each.

How sure are you?

I'm sureI think soI'm guessing

They're the same angle, so the same number of students walk in both charts.

This is a common misconception

Easy trap, because within one chart angle and frequency really do move together. But each chart has its own total. Chart A's 4 students share out 360°, so one student is 90°, and the sector is 1 student. Chart B's 200 students share out the same 360°, so 90° is 50 students. Same angle, very different numbers.

A bigger sector always means more students, even when I'm comparing two different charts.

This is a common misconception

Inside one chart, yes: a bigger sector means a higher frequency. Across charts it breaks. A bigger sector shows a greater proportion of its own data set, not more students. For example, 150 of 200 students walking is a greater proportion than 200 of 600, even though 200 is the bigger frequency.

The angles tell me how many students walk, so I can read the frequencies straight off without anything else.

This is a common misconception

A pie chart shows proportion, not the number of data points. With only the angles and no frequency, you can't work out any frequency. Know any one frequency (the total counts) and you can find the rest: get the multiplier, then divide each angle by it.

Both sectors are the same proportion of their own chart. How many students that is depends on each chart's total, so I need a total or one frequency for each.

This matches the evidence

That's the one. 90° out of 360° is a quarter of each chart, so both show the same proportion; with totals of 4 and 200 that's 1 student and 50 students. And once you know a total or a frequency for each chart, you can compare them. If in School C the walk angle is double the don't-walk angle and 100 students didn't walk, then 200 walked. If half of School D's 380 students walked, that's 190. So 10 more students walked at School C.

A pie chart shows proportion, not frequency, so you need a total or one frequency to say how many. And if you match a chart to a group using real-world sense (a group who revised is more likely to show a larger 'passed' sector), take care: that rests on assumptions.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

A pie chart is a circle cut into sectors; the whole circle is the total frequency, so the angles always add up to 360°.
Have a goMaya has drawn a pie chart with sectors of 100°, 200° and 90°, and she is VERY sure it's perfect. Is she right?
No. Those add up to 390°, and the angles in a pie chart always total 360°.
The whole circle is 360° and stands for the whole data set, so the sectors can't use more than that.
Within one pie chart, the bigger the sector, the higher the frequency.
With pre-drawn equal sectors, a ratio table does the job: 6 students on 12 sectors means a multiplier of 2.
Every angle comes from one multiplier: 360 ÷ total frequency. Multiply each frequency by it.
Have a goA pie chart has a total frequency of 20. Work out the multiplier, then the angle for a frequency of 7.
Multiplier 360 ÷ 20 = 18, so 7 × 18 = 126°.
The multiplier comes from the total, 360 ÷ total, and every frequency is multiplied by that same number.
Never plot the frequencies as angles. Multiply first, then add your angles: they should total 360°.
Awkward multiplier? Keep it as a fraction, round angles to the nearest half degree, and the total may land half a degree from 360°.
Have a goYou calculate a sector angle of 40.7°. What angle do you actually plot with a protractor?
40.5°, the nearest half degree.
A protractor can't plot more precisely than half a degree, and 40.7 is closer to 40.5 than to 41.
To draw it, use a protractor and ruler one sector at a time, lining zero up with the last line you drew.
Going backwards, find the multiplier from a row where you know both frequency and angle, then divide each angle by it.
With angles only and no frequency known, you can't find any frequency. One known frequency, the total counts, unlocks the rest.
A pie chart shows proportion, not frequency, so the same angle on two charts needn't mean the same number.

The big picture

A pie chart splits a 360° circle into sectors in proportion to the frequencies. One multiplier, 360 ÷ total, turns every frequency into its angle, and dividing by it takes you back. A pie chart shows proportion, not frequency, so the same angle on two charts doesn't mean the same number of data points.

Key points

1The whole circle, 360°, stands for the total frequency. Multiplier = 360 ÷ total frequency, and frequency × multiplier = angle.

2Check your angles: they add up to 360° (after rounding to the nearest half degree they may be half a degree out).

3Going backwards, angle ÷ multiplier = frequency, and a missing angle is 360° minus the other angles.

4A pie chart shows proportion, not frequency: with no frequency known you can't find any, but one frequency (or the total) lets you find the rest.

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Worked example

Problem

Chart P has 40 data points and Chart Q has 200. Both have a 45° sector. How many data points does each 45° sector stand for?

⚠ Watch out

Plotting the frequencies as the angles. They must be multiplied by 360 ÷ total first, and the angles must add up to 360°. The cousin of this mistake is thinking the same angle on two different charts means the same number of people.

🧠

Memory hook

Total ↔ 360. Find the multiplier once, 360 ÷ total, and the whole chart follows: multiply to get angles, divide to get frequencies.

✓

Check yourself

Without looking back: a chart has 18 students and one sector stands for 5 of them. Find the multiplier and that angle, and say what all the angles must total.

Flashcards

(15)

What does the whole circle of a pie chart stand for?

The total frequency. All the angles add up to 360°.

What is a sector?

The region between two radii and their connecting arc: one group's slice of the pie chart.

On one pie chart, what does a bigger sector mean?

A higher frequency for that group, within that chart.

How do you find the multiplier from the total frequency?

360 ÷ total frequency. Multiply each frequency by it to get its angle.

How can you check a set of calculated angles?

Add them up. They should come to 360°.

What do you do when the multiplier is not a whole number?

Leave it as a fraction, then round any non-whole angle to the nearest half degree.

After rounding, your angles total 360.5°. Have you gone wrong?

Not necessarily. After rounding to the nearest half degree the total may be half a degree away from 360°.

What are the first moves when drawing a pie chart with a protractor?

Draw a circle, mark its centre and draw a radius. Put the protractor's centre on the circle's centre with its zero on that radius.

How do you set the protractor for the next sector?

Line its zero up with the line you drew most recently.

How do you find a missing angle?

Subtract the other angles from 360°.

You know one row's angle and frequency. How do you find the frequencies of the other sectors?

Multiplier = angle ÷ frequency for that row. Then divide each angle by the multiplier.

150 students are shared over 6 equal sectors. What does one sector stand for?

25, because 150 ÷ 6 = 25.

The angles are shown but no frequency is known. Can you find any frequency?

No. A pie chart only shows proportion. Know any one frequency, the total counts, and you can find the rest.

In a spreadsheet, what does the = at the start of a formula do, and why click a cell instead of typing its number?

The = tells the spreadsheet to start a calculation. Clicking a cell means the output updates automatically when the data changes.

Two pie charts both have a 90° sector. Is it the same frequency?

Not necessarily. Each is the same proportion of its own total: 90° of a chart of 4 data points is 1, but 90° of a chart of 200 is 50.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Where does this working go wrong?

Draw it in the right order

Now run it backwards

Same angle, same number?

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