Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Unit 4, lesson 2) Three angles at a point are 125°, 105° and x°. Find x.
Answer
130. Angles at a point add up to 360°, and 125 + 105 = 230, so x = 360 − 230 = 130. - (Unit 2, lesson 2) Write 3/4 as a percentage.
Answer
75%. 3 ÷ 4 = 0.75, which is 75 hundredths. - (Lesson 3 of this unit) What do the frequencies in a frequency table add up to?
Answer
The number of values that were tallied.
Words used in this lesson
- pie chart
- a circle cut into sectors that show shares of a whole. For example: how 24 students have breakfast.
- sector
- a slice of a circle between two radii. For example: the sector for Rice.
- angle
- the amount of turn of a sector at the centre, in degrees. For example: 150 degrees.
- proportion
- a share of a whole. For example: a quarter of the group.
- total
- the whole amount. For example: 24 students.
- degrees
- the units for angles; a whole circle is 360. For example: 90 degrees is a quarter turn.
Share the circle
A canteen wants one picture of how its 24 students have breakfast. Bars would show the counts, but the canteen wants to show what share of the whole each group takes. A circle is a whole of 360 degrees, so each group can take a sector. Predict first: if 24 students share the 360 degrees equally, how many degrees does one student get? Then set the table slider to 24 students in the figure and check, and try the other tables.
One student is 360 ÷ 36 = 10°. The Walk sector holds 15 students, 15 × 10 = 150°.
Show me the steps
- 24 students: one student is 360 ÷ 24 = 15°. Groups of 10, 6, 5 and 3 give 150°, 90°, 75° and 45°.
- 36 students: one student is 10°. Groups of 15, 11, 6 and 4 give 150°, 110°, 60° and 40°.
- 40 students: one student is 9°. Groups of 14, 12, 10 and 4 give 126°, 108°, 90° and 36°.
- 72 students: one student is 5°. Groups of 30, 24, 12 and 6 give 150°, 120°, 60° and 30°. Every table's angles add to 360°.
The angle of a sector is its share of 360°. Angle = frequency ÷ total × 360°. Often it is quicker to find one item first: 360 ÷ total is the angle for one, and the sector is the frequency times that.
Why the angles add to 360°: the frequencies add to the total, and the shares of one whole add to one whole circle. If your angles do not add to 360°, check each one.
From a percentage
Sometimes the data comes as percentages. A percentage is a share out of 100, so it is that fraction of 360°: 10% is a tenth of 360°, which is 36°. Move the slider and read the angle.
45% of the circle is 45 ÷ 100 × 360 = 162°. The rest is 55%, which is 198°.
Show me the steps
- 10%: 10 ÷ 100 × 360 = 36°. The rest is 90%, which is 324°.
- 25%: a quarter of 360 is 90°. The rest is 270°.
- 50%: half the circle is 180°, and so is the rest.
- 75%: three quarters is 270°, and the rest is 90°. Each step of 5% moves the sector on by 18°.
From a percentage: angle = percentage ÷ 100 × 360°. Each 1% is 3.6°, so each 10% is 36° and each 5% is 18°.
A pie chart shows proportion, not amount. Two pies can have the same sector angle and stand for very different numbers of people.
A common mistake: “The angle is the number of students.” In a pie of 24 students a group of 10 is not 10°. Each student is 15°, so the group is 10 × 15 = 150°.
Worked example: angles from a table
50 students choose a club: Reading 20, Games 15, Sport 10, Music 5. Find each sector angle.
- Reading: 20 ÷ 50 = 0.4, and 0.4 × 360 = 144°.
- Games: 15 ÷ 50 = 0.3, and 0.3 × 360 = 108°. Sport: 10 ÷ 50 = 0.2, and 0.2 × 360 = 72°. Music: 5 ÷ 50 = 0.1, and 0.1 × 360 = 36°.
- Check: 144 + 108 + 72 + 36 = 360. 144°, 108°, 72° and 36°.
- Draw it: make a circle and one radius. Measure 144° from that radius with a protractor and draw the next radius. From that new line measure 108°, then 72°. The last sector should be 36° when you check it. Label each sector or add a key.
Your go. The first step is given. 15 pupils choose a pet: Cats 6, Dogs 5, Fish 3, Birds 1. Find each sector angle.
- One pupil is 360 ÷ 15 = 24°.
- Now find the four angles yourself and check that they add to 360°, then check.
Check the answer
Cats 6 × 24 = 144°, Dogs 5 × 24 = 120°, Fish 3 × 24 = 72°, Birds 1 × 24 = 24°. Check: 144 + 120 + 72 + 24 = 360.
Check before you go on
1. In a survey of 30 students, 6 chose pizza. What is the angle of the pizza sector in a pie chart?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A pie chart is made from 20 students. What angle in degrees stands for one student?
3. In a pie chart of 18 students, 5 prefer swimming. What is the angle of the swimming sector, in degrees?
Level 2 · Building
4. 30% of the students in a survey prefer apples. What angle does the apple sector have in a pie chart, in degrees?
5. In a pie chart of 25 students, one sector has an angle of 72°. How many students does it stand for?
6. A pie chart has four sectors. Three of them have angles 140°, 95° and 65°. What is the angle of the fourth sector, in degrees?
Level 3 · Stretch
7. Chart A is a pie chart for 40 students and chart B for 200 students. In both, the Bus sector is 90°. Which statement is true?
8. A pie chart shows 120 tickets sold. The Adult sector is 150°, the Child sector 90° and the Senior sector is the rest. How many Senior tickets were sold?
Something to think about
A pie chart for 24 students has a sector of 90 degrees. A pie chart for 200 students has a sector of 90 degrees as well. A student says, “They are the same size, so the same number of students.” What would you say?
One way to think about it
Both sectors are a quarter of their circle, which is the same share. A quarter of 24 is 6 students and a quarter of 200 is 50 students. A pie chart shows the share of its own whole, so you need the total to turn an angle into a number.
Exit ticket
9. A pie chart of 60 students has a 120° sector for Bus. Which statement is true?