A whole, cut into shares
A class records how each child comes to school. Half of the children walk, a quarter take the bus and a quarter come by car. A pie chart shows this as a circle cut into slices, called sectors, one for each way of traveling. The whole circle stands for the whole class, and each sector is the fraction of the class that gave that answer.
The walk sector is half of the circle, . The bus sector and the car sector are a quarter each, .
The walk sector fills half of the circle; the bus and car sectors fill a quarter each.
The sectors make one whole
Every child comes to school one way, so every child is in exactly one sector. The sectors together are the whole class: , with nothing left over and nothing counted twice.
That is what a pie chart needs: categories that share out one whole. A survey in which one person can give two answers, such as “Which sports do you play?”, cannot be drawn as a pie chart, because its parts would add up to more than the whole.
A pie chart shows shares, not sizes. This one says that half of the class walks, but not how many children that is: half of a class of 30 is 15, and half of a school of 400 is 200, and the two pie charts look exactly the same. To compare the sizes of categories, a bar chart is the better choice.
Fractions, percentages and angles
The whole class is 1, which is 100%. The whole circle is one full turn about its center, which is 360°. So every fraction of the class is the same fraction of 100% and of 360°.
The bus sector is of the class, so it is % = 25% of the children, and the angle at the center of its sector is , a right angle. The walk sector is , a straight line through the center. The angles add up to a full turn: 180° + 90° + 90° = 360°.
From counts to angles
Survey results usually arrive as counts. Twenty-four students name their favorite fruit: 9 choose apple, 6 banana, 4 grape and 5 orange. Apple is 9 out of 24, so its sector is of the circle, and its angle is .
A quicker way is to find the angle for one student first. The 24 students share 360°, so each student gets 360° ÷ 24 = 15°. Then apple is 9 × 15° = 135°, banana is 6 × 15° = 90°, grape is 4 × 15° = 60° and orange is 5 × 15° = 75°. Check that they fill the circle: 135° + 90° + 60° + 75° = 360°.
To draw the chart, draw a circle and one radius. Measure the first angle from that radius with a protractor and draw the second radius; measure the next angle from the second radius, and so on around the circle. The last sector is what is left, and it should measure 75°. Label each sector with its category.
Each of the 24 students has 15° of the circle: apple 135°, banana 90°, grape 60° and orange 75°, which add up to 360°.
From an angle back to a count
Reading a pie chart works the other way. A sector’s angle out of 360° is the fraction of the whole it stands for. In a pie chart of 60 people, a sector of 72° is of the circle, so it stands for people, which is 20% of them.
The number of people a sector stands for depends on the total. The same 72° sector in a pie chart of 300 people stands for people.
Three slips
Do not use the counts as the angles. Drawing apple as 9°, banana as 6° and so on gives sectors that add up to only 24°, a thin wedge of the circle. Each count must first be turned into a fraction of 360°.
A percentage is not an angle. A sector that is 25% of the pie is not 25°: it is a quarter of a full turn, 90°.
The angle of a sector depends on how many sectors share the circle. A pie cut into 5 equal sectors has 360° ÷ 5 = 72° in each, so 2 of them cover 2 × 72° = 144°, which is % of the pie. The angle is not 40°, which is the percentage, and not 2 × 10° = 20°, which treats every sector as 10° however many there are.
Worked example: A Pie Chart Drawn from a Travel Survey, and a Sector Read Back to a Count
Question A school asked its 240 Secondary 2 students how they travel to school. 96 take the bus, 72 take the train, 48 walk and 24 come by car. The results are to be drawn as a pie chart. (a) Find the angle of each sector. (b) A second school has 450 students and draws the same kind of pie chart. Its sector for the students who walk has an angle of 56°. How many students of the second school walk?
1.Find the angle for one student. The 240 students share 360° equally, so each student has 360 ÷ 240 = 1.5°.
The 240 students share the 360° of the circle, so each student has 360 ÷ 240 = 1.5°. 2.Multiply each count by 1.5°. Bus: 96 × 1.5 = 144°. Train: 72 × 1.5 = 108°. Walking: 48 × 1.5 = 72°. Car: 24 × 1.5 = 36°.
Each count times 1.5° is the angle of its sector: 144°, 108°, 72° and 36°. 3.(a) The angles are 144°, 108°, 72° and 36°. Check: 144 + 108 + 72 + 36 = 360, so the four sectors fill the circle exactly.
(a) 144 + 108 + 72 + 36 = 360, so the four sectors fill the circle. 4.For the second school, write the sector as a fraction of the circle: 56360 = 745. The students who walk are the same fraction of the 450 students.
In the second school the sector for walking is 56360 = 745 of the circle. 5.(b) 745 × 450 = 70 students walk. Check: in the second school each student has 360 ÷ 450 = 0.8°, and 70 × 0.8 = 56°.
(b) 745 × 450 = 70 students walk.
Answer: (a) bus 144°, train 108°, walking 72°, car 36°, which add up to 360°; (b) 70 students
Common mistakes
- Using the counts as the angles, so that the bus sector is drawn as 96°. The four counts add up to 240, not 360, so the sectors would not fill the circle. Each count has to be scaled by 360240 first.
- Reading the 56° sector with the first school's 1.5° for each student. The angle for one student depends on the size of the survey. The second school has 450 students, so its sectors are fractions of 450.
Worked example: Pie Chart with Angle, Percentage, and Count Hybrids
Question A pie chart represents the enrollment of 360 students across four school activities: Sports, Arts, Robotics, and Uniformed Groups. The sector for Sports has an angle of 120°. The sector for Arts represents 25% of the total students. There are 45 students in Robotics, and the remaining students are in Uniformed Groups. (a) What was the sector angle representing Uniformed Groups on the pie chart? (b) How many students were in Uniformed Groups?
1.Total circle = 360° = 360 students. Therefore, 1° = 1 student.
360 students share 360°, so one student is one degree. 2.Sports: 120° = 120 students.
Sports: 120°, 120 students. 3.Arts: 25% = 14 × 360 = 90 students = 90°.
Arts: a quarter, 90 students, 90°. 4.Robotics: 45 students = 45°.
Robotics: 45 students, 45°. 5.Uniformed Groups students: 360 − (120 + 90 + 45) = 360 − 255 = 105 students.
(b) The rest: 360 − 255 = 105 students. 6.(b) 105 students.
7.Since 1 student = 1°, angle is 105°.
(a) 105 students is 105°. 8.(a) 105°.
Answer: (a) 105°; (b) 105 students
Common mistakes
- Adding percentage (25) directly to degrees (120) without converting 25% to 90°.
- Assuming Uniformed Groups is 100° because 360 minus 255 is miscalculated.