Math 6

Circle graphs

A whole group splits into a few parts. How can one picture show at a glance what share of the whole each part is?

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Finding a percent of a quantityMath 6 DegreesGrade 4

Where you are 12 in Data displays

Statistics7

Data displays12

▴ 7 above Choosing a display for a question Misleading graphs Box plots3 Two-way tables3

Circle graphs

When you reach for it

Some questions about data ask “what share?” rather than “how many?” What part of all the tickets went to adults? Does sleep take more or less than a third of a day? A circle graph, also called a pie chart, answers that kind of question with a picture: the whole circle stands for the whole group, and each category gets a slice whose size shows its share.

It suits categorical data that split one whole into a few parts, with every item in exactly one category. Reading a circle graph means turning a slice back into a fraction or a percent, and then into a count. Making one goes the other way: from counts to shares to angles at the center.

The procedure

What a slice means

A sector is the part of a circle between two radii, shaped like a slice of pie. In a circle graph the sectors fill the circle without overlapping, one per category, and the rule that makes the picture honest is this: each sector’s share of the circle equals its category’s share of the data.

The share of the circle is measured by the angle at the center. A full turn is 360^\circ (Degrees), so a sector whose angle is a degrees is \displaystyle \frac{a}{360} of the circle. Its area is that same fraction of the circle’s area, which is why the eye can judge it; sectors and their areas makes that exact later on.

So one share can be named three ways, and they are always equal:

\text{share} = \frac{\text{count}}{\text{total}} = \frac{\text{percent}}{100} = \frac{\text{angle}}{360^\circ}

The whole data set is 100\% and gets all 360^\circ, so each 1\% gets 360 \div 100 = 3.6^\circ. A few shares are worth knowing by sight:

Share	Percent	Angle at the center
the whole	100\%	360^\circ
a half	50\%	180^\circ
a third	about 33.3\%	120^\circ
a quarter	25\%	90^\circ, a right angle
a tenth	10\%	36^\circ

Reading a circle graph

A circle graph labeled with percents gives each share directly. To turn a share into a count you need the total, which a well-made graph states in its title. The count is then that percent of the total, which is finding a percent of a number. A graph labeled with angles gives the share as \displaystyle \frac{\text{angle}}{360}, and the count is that fraction of the total. With no numbers printed at all, the best you can do is compare each slice with a half, a quarter or a tenth, which gives an estimate and no more.

Making a circle graph

Going the other way, each category’s share is its count over the total (Finding what percent one number is of another). Write the share as a percent, then take that share of 360^\circ for the angle. Two checks come free: the percents must add to 100\% and the angles to 360^\circ, because the shares are the pieces of one whole.

To draw it, draw a circle and one radius straight up. Lay a protractor (Measuring and drawing angles with a protractor) with its center on the circle’s center and its zero line along that radius, mark the first angle, and draw the second radius. The next sector is measured from that new radius, and so on clockwise. The last sector is whatever is left, and measuring it checks the whole drawing. Label every sector with its category and its percent, and give the graph a title and the total.

When the shares are not whole numbers, the angles are rounded to whole degrees, since that is what a protractor marks, and the percents to a whole number or a tenth. Rounded parts can add to 361^\circ or 99\%. The total is fixed, so one part gives back or takes the extra unit. Which part is a choice, not a rule; this lesson picks the part rounding moved furthest, the closest call. Example 4 does this.

Worked examples

Example 1

Turning percents into counts

Gavril asked 60 students at his school to name their favorite sport, and each named one.

Figure 1Gavril’s poll of 60 students. Basketball’s sector is a quarter of the circle, so its two radii meet at a right angle.

(a) How many students chose swimming? (b) How many more chose soccer than basketball? (c) What fraction of the students chose soccer or basketball, and how many students is that?

Solution

The graph gives percents and the total is 60, so each count is a percent of 60.

(a) Swimming is 20\%: 0.2 \times 60 = 12 students.

(b) Soccer is 0.35 \times 60 = 21 students and basketball is 0.25 \times 60 = 15, so soccer has 21 - 15 = 6 more. A shorter route subtracts first: 35\% - 25\% = 10\%, and 10\% of 60 is 6. Subtracting percents is allowed here only because both are percents of the same 60.

(c) Together they are 35\% + 25\% = 60\%, which is \displaystyle \frac{60}{100} = \frac{3}{5} of the students, and \displaystyle \frac{3}{5} \times 60 = 36 students. On the graph the two sectors together cover a little more than half the circle, as 60\% should.

Example 2

Reading angles instead of percents

Gemma’s school sold 240 tickets for its play. The graph labels each sector with its angle at the center.

Figure 2The 240 tickets for the school play, each sector labeled with its angle at the center.

How many tickets of each kind were sold, and what percent were adult tickets?

Solution

A sector of a degrees is \displaystyle \frac{a}{360} of the circle, so it stands for \displaystyle \frac{a}{360} of the 240 tickets. Simplifying the fraction first keeps the numbers small.

Ticket	Angle	Share of the circle	Tickets
Adult	150^\circ	\displaystyle \frac{150}{360} = \frac{5}{12}	\displaystyle \frac{5}{12} \times 240 = 100
Student	120^\circ	\displaystyle \frac{120}{360} = \frac{1}{3}	\displaystyle \frac{1}{3} \times 240 = 80
Child	60^\circ	\displaystyle \frac{60}{360} = \frac{1}{6}	\displaystyle \frac{1}{6} \times 240 = 40
Staff	30^\circ	\displaystyle \frac{30}{360} = \frac{1}{12}	\displaystyle \frac{1}{12} \times 240 = 20

Check: 100 + 80 + 40 + 20 = 240. Another route gives the same: each degree stands for \displaystyle \frac{240}{360} = \frac{2}{3} of a ticket, and \displaystyle 150 \times \frac{2}{3} = 100.

Adult tickets were \displaystyle \frac{5}{12} of the total, and 5 \div 12 \approx 0.4167, so about 41.7\% (Converting among fractions, decimals and percents).

Example 3

Making a circle graph from counts

Gesa kept track of one school day, all 24 hours of it: sleep 9 hours, school 7, free time 4, meals 2 and homework 2. Make a circle graph.

Solution

Every hour of the day sits in exactly one activity, and 9 + 7 + 4 + 2 + 2 = 24, so these are parts of one whole and a circle graph fits. Each share is hours over 24, and the angle is that share of 360^\circ. Since 360 \div 24 = 15, every hour gets 15^\circ.

Activity	Hours	Percent	Angle
Sleep	9	37.5\%	135^\circ
School	7	about 29.2\%	105^\circ
Free time	4	about 16.7\%	60^\circ
Meals	2	about 8.3\%	30^\circ
Homework	2	about 8.3\%	30^\circ

Both checks pass: 135 + 105 + 60 + 30 + 30 = 360, and the rounded percents give 37.5 + 29.2 + 16.7 + 8.3 + 8.3 = 100.

Draw the radius straight up, measure 135^\circ clockwise for sleep, then 105^\circ from the new radius for school, then 60^\circ and 30^\circ. The 30^\circ left over is homework, and measuring it checks the drawing.

Figure 3Gesa’s day of 24 hours, drawn at 15^\circ an hour and labeled with rounded percents.

Example 4

Rounding so the parts still add up

Giorgos asked the 26 students in his class to name their favorite sport: running 9, cycling 8, swimming 6 and climbing 3. Find each angle to the nearest degree and each percent to the nearest whole percent.

Solution

Each angle is \displaystyle \frac{\text{count}}{26} \times 360 and each percent \displaystyle \frac{\text{count}}{26} \times 100.

Sport	Count	Exact angle	Rounded
Running	9	124.62	125
Cycling	8	110.77	111
Swimming	6	83.08	83
Climbing	3	41.54	42
Total	26	360	361

The exact angles add to 360, but rounding pushed three of them up and the total to 361^\circ. That graph cannot be drawn: the last sector would overlap the first by a degree. One part has to give the degree back, and this lesson picks the part rounding moved furthest. Climbing went up by 42 - 41.54 = 0.46, more than running (0.38) or cycling (0.23), so climbing gets 41^\circ:

125 + 111 + 83 + 41 = 360

The exact percents, 34.62, 30.77, 23.08 and 11.54, drift the same way, to 35 + 31 + 23 + 12 = 101. Climbing again moved furthest, so it gets 11\% and 35 + 31 + 23 + 11 = 100. Every part is still within 1 of its exact value, and the parts make the whole.

Example 5

Two wholes of different sizes

A choir gave two concerts. Saturday’s sold 400 tickets and Sunday’s sold 900. Two circle graphs of the same size show the kinds of ticket sold: the student sector is 30\% on Saturday’s graph and 20\% on Sunday’s. (a) Which concert sold more student tickets? (b) What percent of all the weekend’s tickets were student tickets?

Solution

(a) Saturday sold 0.3 \times 400 = 120 student tickets and Sunday 0.2 \times 900 = 180. Sunday sold more, although its sector is smaller. Each circle stands for its own whole, so a sector can be compared only with sectors in the same circle. Drawing both circles the same size hides that Sunday’s whole is more than twice Saturday’s.

(b) Not the average of 30\% and 20\%. Count instead: 120 + 180 = 300 student tickets out of 400 + 900 = 1{,}300, and

\frac{300}{1{,}300} = \frac{3}{13} \approx 23.1\%

The answer sits nearer 20\% because Sunday’s larger concert carries more of the tickets. Comparing groups of different sizes needs the counts, or a two-way table.

Example 6

From one sector to the whole

Gilles’s graph shows the favorite sports of the students in his sports club, labeled with angles. The key says the swimming sector stands for 18 students, but the total is missing.

Figure 4Gilles’s sports club. The swimming sector stands for 18 students.

How many students are in the club, and how many chose each of the other sports?

Solution

Swimming’s sector is 72^\circ, and \displaystyle \frac{72}{360} = \frac{1}{5} of the circle. So 18 students are one fifth of the club, and the club has 18 \times 5 = 90 students (Finding the whole from a part and a percent). Each other sector is its share of 90:

\begin{aligned} \text{Soccer: } & \tfrac{108}{360} \times 90 = 27 \\ \text{Basketball: } & \tfrac{100}{360} \times 90 = 25 \\ \text{Other: } & \tfrac{80}{360} \times 90 = 20 \end{aligned}

Check: 27 + 25 + 18 + 20 = 90. Put another way, each student gets 360 \div 90 = 4 degrees, and swimming’s 72^\circ is 72 \div 4 = 18 students, as the key says.

Boundaries

The parts must make one whole. Every item belongs to exactly one category and none is left out, so the percents add to 100\% and the angles to 360^\circ. A poll that lets each person choose several answers gives percents that add to more than 100\%; a table with a category missing gives less. Neither can be drawn as a circle graph. A bar graph of the percents can show them (Bar graphs and pictographs).

Categories, not numbers on a scale. Temperatures, heights and race times are numerical data, and their values do not add up to a whole. Gesa’s day worked because its categories are activities and the hours are how much of the day each took.

A handful of sectors. Many thin sectors are hard to tell apart and hard to label. A bar graph compares them more easily, or the smallest categories can be gathered into one “other” sector.

Shares, not sizes. A circle graph shows each category’s share of its own whole. Counts need the total, and two circle graphs for different totals cannot be compared sector by sector, as Example 5 showed.

Printed values are rounded. A count worked out from a rounded percent may come out a little off, or not a whole number; the nearest whole number that fits the data is the count.

Where this goes next

Which display answers which question, a circle graph, a bar graph or a dot plot, is the subject of Choosing a display for a question, and the ways a pie can be drawn to mislead, tilted or exploded, belong to Misleading graphs. The same shares written in a table are a relative frequency table. Later, Sectors, segments and their areas takes the fraction \displaystyle \frac{a}{360} of this lesson and applies it to a circle’s area and its edge, so a slice’s area and the length of its curved side can be worked out exactly. A spinner is a circle graph turned into a game: the chance of landing in a sector is its share of the turn (Probability models: equally likely or not).

Exercises

Reading a circle graph

Exercises 1 to 4 use this graph of where the 800 tickets for a concert were sold.

Figure 5Where the 800 tickets for a concert were sold.

What percent of the tickets were sold online, and how many tickets is that?
How many more tickets were sold at the box office than on the door?
What fraction of the tickets were not sold online? Give it in simplest form, and find how many tickets that is.
What is the angle at the center of the box office sector?

Reading angles

A circle graph of how Gemma spent all 24 hours of one Saturday has a sleep sector of 150^\circ and a reading sector of 45^\circ. How many hours did she sleep? What fraction of the day did she spend reading, and how many hours is that?
A circle graph shows the favorite sports of 72 students. The soccer sector measures 140^\circ and the hockey sector 25^\circ. How many students chose each?

Making a circle graph

Gilles asked 40 students to name their favorite sport: soccer 12, basketball 10, swimming 8, tennis 6 and other 4. Find the percent and the angle for each sector, and check both totals.
A school play sold 180 tickets: 81 adult, 54 student and 45 child. Find the percent and the angle for each kind of ticket.
Gavril’s school day of 24 hours: sleep 10 hours, school 7, sport 2, meals 1.5 and other 3.5. Find the angle of each sector.
Of 21 students, 10 named soccer as their favorite sport, 6 netball and 5 hockey. Find each percent to the nearest whole percent so that they add to 100\%, then each angle to the nearest degree.
Of 18 students, 8 named running, 6 swimming and 4 cycling. Find each percent to the nearest whole percent so that they add to 100\%.

Does a circle graph fit?

A poll asked 50 students to tick every sport they play: 60\% ticked soccer, 45\% swimming and 30\% tennis. Can one circle graph show these? Explain.
Would a circle graph suit each of these? Explain. (a) The noon temperature on each day of one week. (b) The favorite sports of 30 students, who named 15 different sports between them.
Two circle graphs of the same size show the tickets sold for two concerts. One concert sold 2{,}000 tickets and its child sector is 10\%; the other sold 400 and its child sector is 25\%. Which concert sold more child tickets?
A circle graph of a class’s favorite sports has three sectors, labeled 40\%, 35\% and 20\%. What is wrong with it?

Working backwards

In a circle graph of ticket sales, a 54^\circ sector stands for 12 tickets. How many tickets were sold in all?
Three of the four sectors of a circle graph measure 130^\circ, 95^\circ and 75^\circ. Find the angle of the fourth. If the graph shows 144 students, how many does the fourth sector stand for?

Reasoning

Explain why the angles of a circle graph always add to 360^\circ, and its percents to 100\%.
Explain why each 1\% of the data gets 3.6^\circ of the circle.
In one circle graph, sector A has twice the angle of sector B. Explain why A stands for twice as many items as B. Then explain why that reasoning fails when A and B are in two different circle graphs.

Answers

45\%, which is 0.45 \times 800 = 360 tickets.
160 more. Box office 0.3 \times 800 = 240, on the door 0.1 \times 800 = 80, and 240 - 80 = 160. Or: 30\% - 10\% = 20\%, and 20\% of 800 is 160.
\displaystyle \frac{11}{20}, which is 440 tickets. Not online is 100\% - 45\% = 55\%, and \displaystyle \frac{55}{100} = \frac{11}{20}; then \displaystyle \frac{11}{20} \times 800 = 440.
108^\circ. The sector is 30\% of the circle, and 0.3 \times 360 = 108.
Sleep 10 hours, since \displaystyle \frac{150}{360} \times 24 = 10. Reading took \displaystyle \frac{45}{360} = \frac{1}{8} of the day, and \displaystyle \frac{1}{8} \times 24 = 3 hours.
Soccer 28, hockey 5. Each degree stands for \displaystyle \frac{72}{360} = \frac{1}{5} of a student, so \displaystyle 140 \times \frac{1}{5} = 28 and \displaystyle 25 \times \frac{1}{5} = 5.
Soccer 30\%, 108^\circ; basketball 25\%, 90^\circ; swimming 20\%, 72^\circ; tennis 15\%, 54^\circ; other 10\%, 36^\circ. Each student is \displaystyle \frac{1}{40} = 2.5\% of the poll and 9^\circ of the circle. Totals: 100\% and 360^\circ.
Adult 45\%, 162^\circ; student 30\%, 108^\circ; child 25\%, 90^\circ. For example \displaystyle \frac{81}{180} = 0.45 and 0.45 \times 360 = 162. The totals are 100\% and 360^\circ.
Sleep 150^\circ, school 105^\circ, sport 30^\circ, meals 22.5^\circ, other 52.5^\circ. Each hour gets 360 \div 24 = 15^\circ, and the angles add to 360^\circ.
Soccer 48\%, netball 28\%, hockey 24\%; angles 171^\circ, 103^\circ, 86^\circ. The exact percents 47.62, 28.57 and 23.81 round to 48 + 29 + 24 = 101; netball moved furthest (up 0.43), so it gets 28\%. The exact angles 171.43, 102.86 and 85.71 round to 171 + 103 + 86 = 360, so no angle needs changing.
Running 45\%, swimming 33\%, cycling 22\%. The exact percents 44.44, 33.33 and 22.22 round to 44 + 33 + 22 = 99; running moved furthest (down 0.44), so it gets 45\%.
No. The percents add to 60 + 45 + 30 = 135\%, more than the whole. A student who plays two of the sports is counted in two sectors, so the sectors are not parts of one whole. A bar graph of the three percents would work.
(a) No. Temperatures are numbers on a scale, not categories, and adding them gives no whole; a line graph suits them. (b) Poorly. Fifteen sectors, most of them thin, are hard to compare and label; a bar graph is clearer, or the rare sports can be gathered into one “other” sector.
The concert with 2{,}000 tickets: 0.1 \times 2{,}000 = 200 child tickets, against 0.25 \times 400 = 100. The smaller sector is the larger count because its whole is five times bigger.
The percents add to 40 + 35 + 20 = 95\%, not 100\%. Some 5\% of the class, 18^\circ of the circle, is not shown: a category was left out, or some students did not answer. That part has to be added before the graph shows one whole.
80 tickets. The sector is \displaystyle \frac{54}{360} = \frac{3}{20} of the circle, so 12 tickets are \displaystyle \frac{3}{20} of the total: \displaystyle \frac{1}{20} is 4 tickets and the total is 20 \times 4 = 80.
60^\circ, and 24 students. 360 - (130 + 95 + 75) = 60, and \displaystyle \frac{60}{360} \times 144 = 24.
The sectors fill the circle with no gaps and no overlaps, so together their angles make one full turn, 360^\circ. They stand for the categories, and each item is in exactly one category, so the shares together are the whole data set, 100\%.
The whole data set, 100\%, gets the whole circle, 360^\circ. Each share of the circle equals its share of the data, so 1\%, one hundredth of the data, gets one hundredth of the circle: 360 \div 100 = 3.6^\circ.
In one graph a sector stands for \displaystyle \frac{\text{angle}}{360} of the same total, so twice the angle gives twice the share of the same number, and twice as many items. Two graphs have their own totals: twice the angle of a smaller total can be fewer items. In Exercise 14, 25\% of 400 is 100, while 10\% of 2{,}000 is 200.