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This page sets out the method for pie charts in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Pie charts topic page.
When you see a frequency table and 'draw a pie chart', find how many degrees each single item is worth.
When you see 'work out the angle for …' for a single category, use frequency over total.
The angle is the fraction of the total, times 360°.
10 out of 40 is 1/4, and 1/4 of 360° is 90°.
Divide the angle by 360 to get the fraction, then multiply by the total.
90° is 1/4 of the people.
The angles must add to 360°.
If they do not, a category is missing or an angle is wrong.
Find the total frequency before you divide.
Then each category is its frequency over that total.
A sector is one slice.
Its angle is what you draw with a protractor.
Percentages are out of 100. Pie charts are out of 360.
A 25% slice is 90°, not 25°.
A circle is 360^°, not 100^°. Percentages are out of 100, and a pie chart is out of 360. A quarter of the people is 25 percent, and it is also 90^°. Writing 25 as the angle mixes the two systems. The protractor measures degrees, so the calculation has to end in degrees.
Choose totals that split 360 evenly when you are drawing your own chart, such as 12, 30 or 90. Then each person, or each small group, is worth a whole number of degrees. 360 ÷ 30 = 12, so each person in a survey of 30 is worth 12^°. That check stops a lot of arithmetic slips.
Method: 1. Remember that the whole pie is 360 degrees. 2. Do not multiply the fraction by 100. That makes a percentage, not an angle. 3. Remember that a quarter turn is 90 degrees and a half turn is 180 degrees. 4. If the total divides 360, each item is worth 360 divided by the total, in degrees.
Write the category as a fraction of the total before you touch 360. Ten out of thirty is 10/30. Simplify if it helps: 1/3. Then multiply by 360. One third of 360 is 120, so the slice is 120^°. Simplifying is optional. 10/30 × 360 gives the same 120.
The denominator is the grand total. If three categories are 10, 8 and 12, the total is 30, not 10. Using the category you are drawing as the denominator makes a fraction bigger than the truth, and the angle comes out too large. Add every category first if the question has not already given the total.
Method: 1. Add all the frequencies if the total is not given. 2. Write frequency over total for the category you need. 3. Multiply that fraction by 360. 4. Check the fraction is less than 1, so the angle is less than 360.
Sometimes the pie is already drawn and you are given an angle. The angle is the fraction dfracangle360 of the people. Multiply that fraction by the total. An angle of 120^° in a survey of 90 people is 120/360 × 90. That is one third of 90, which is 30 people.
A useful shortcut is to find how many degrees one person is worth, then divide the angle by that. In a survey of 90, each person is 360 ÷ 90 = 4 degrees. An angle of 120 degrees is 120 ÷ 4 = 30 people. Both methods must agree. If they do not, the total or the angle has been misread.
Method: 1. Divide the given angle by 360. That is the fraction of the pie. 2. Multiply the fraction by the total frequency. 3. Or divide 360 by the total to find the degrees for one item. 4. Divide the angle by the degrees for one item to recover the frequency.
The angles of one pie add to 360^°. Calculate every category, then add. This catches a wrong total. It also catches a missing group, such as the people who said "other". If three angles add to 300, the missing slice is 60^°, and that slice belongs to the category you have not drawn yet.
A missing frequency is the same idea in people rather than degrees. Subtract the known frequencies from the total, then turn that remainder into an angle. Do not draw the remainder as whatever space is left on the page. Measure the angle you calculated. The eye is a poor protractor.
Method: 1. Add the angles you have calculated. 2. Add the angles and correct them if they are not 360 degrees. 3. Find a missing category by subtracting the known frequencies from the total. 4. Turn that missing frequency into an angle with the same total as the other slices.
The quickest way to build or read a pie chart is to find what one item is worth in degrees. Divide 360 by the total frequency. For 36 pupils, 360 ÷ 36 = 10, so each pupil is 10^°. Then multiply by the frequency of each category: 15 pupils is 15 × 10 = 150^°. This avoids long fractions and gives angles you can check at a glance.
Some totals do not divide nicely into 360, such as 45 or 24. They still work. For 45 people each person is 360 ÷ 45 = 8^°, and for 24 people each is 15^°. If the total does not divide into 360 exactly, use the fraction method: the frequency over the total, multiplied by 360. Either way, add all your angles at the end. They must total 360^°.
Method: 1. Add the frequencies to find the total, if it is not given. 2. Divide 360 by the total to find the degrees for one item. 3. Multiply each frequency by that number to get its angle. 4. Add all the angles and check they make 360.
Tip: Write the degrees for one item at the top of your working. Every angle then becomes a single multiplication.
Angles on a pie chart always add to 360^°, so a missing angle is 360 minus the angles you know. If two sectors are given as 150^° and 90^°, the others share 360 - 150 - 90 = 120^°. This step is worth a method mark by itself, so write it as a line of working.
A clue such as "classical has twice as many votes as jazz" means that the two angles are in the ratio 2 : 1. The 120^° is split into 3 equal parts of 40^°, so jazz is 40^° and classical is 80^°. Check that 40 + 80 = 120. Sectors for equal numbers have equal angles, so "the same number" means "the same angle".
Method: 1. Subtract the known angles from 360. 2. Turn the clue into parts, such as 2 parts and 1 part. 3. Divide the remaining angle by the total parts to find one part. 4. Multiply by the number of parts for each category and check the sum.
To go from an angle to a number of people, use the fraction dfracangle360 of the total. A sector of 45^° in a chart of 120 people is 45/360 × 120 = 15 people. Simplify first if you can: 45/360 = 1/8, and one eighth of 120 is 15. This is easier without a calculator.
Sometimes the total is not given. A typical question says that "there were 75 more people on Friday than on Thursday" and asks for the number on Saturday. The difference in the two angles matches the difference in the number of people. If Friday is 130^° and Thursday is 80^°, then 50^° stands for 75 people, so 1^° is 1.5 people. Saturday at 150^° is 150 × 1.5 = 225 people.
Method: 1. Decide whether the total is given. If it is, use angle over 360 times the total. 2. If it is not, find the difference in two angles and the difference in the matching numbers of people. 3. Divide to find the number of people for one degree. 4. Multiply by the angle you need and state the answer in people.
A pie chart shows proportions, not totals. Two pie charts with the same size sector do not show the same number of people unless the groups are the same size. School A has 200 pupils and School B has 80 pupils. If the cat sector is 90^° in A and 135^° in B, then A has 90/360 × 200 = 50 cat lovers and B has 135/360 × 80 = 30. The bigger sector is in the smaller school.
When you are asked what is wrong with a pie chart, check that the angles add to 360^°, that every sector is labelled or has a key, that the chart has a title, and that the sizes match the data. A comparison needs both totals. Without them you can only compare the share of each group, and you should say so using the words "a greater proportion".
Method: 1. Find the total for each chart. 2. Convert each angle into a number of people. 3. Compare the numbers, or compare the proportions if the totals are not known. 4. State the answer in the context of the question.
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