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Bar Graphs and Circle Graphs

Read bar, double bar, and circle graphs and use their numbers to find differences, totals, percents, central angles, and missing values.

Practice this skill 25 min Bar graphs and circle graphs Leer en español
Key idea

What you'll learn

Much of the information you see every day comes as a picture: a bar graph of sales on the news, a pie chart of where your tax dollars go, a chart from your bank showing your spending. In this lesson you will learn to read these graphs accurately and to use the numbers in them to answer questions.

You will learn to:

  • read the height of a bar against the scale, even when it falls between gridlines,
  • compare bars in a single or a double bar graph,
  • turn the percents in a circle graph into real amounts (dollars, people, hours),
  • find the central angle of a circle-graph section,
  • work backwards from a section to the total, and
  • turn bar-graph data into circle-graph data.

Why it matters on the GED: Data questions make up a real share of the Mathematical Reasoning test, and many of them start with a bar graph or a circle graph. The math is usually simple — subtract, multiply by a percent, divide — but the test checks whether you read the graph carefully. Most wrong answers come from reading the wrong bar, mixing up a count with a percent, or forgetting what the whole is. Slow, careful reading wins these points.

Words to know

  • Categorical data — data that sorts things into groups, not numbers (car, bus, train; or rent, food, savings).
  • Bar graph — a graph that uses bars to show the count or amount in each category. Taller bar = bigger amount.
  • Scale — the numbers along the side (axis) of a bar graph. A scale might count by 1s, 5s, 10s, or 1,000s.
  • Gridline — a faint line across the graph at each scale number, to help you read the bar height.
  • Double bar graph — a bar graph with two bars for each category, so you can compare two groups (Store A vs. Store B).
  • Legend (key) — the box that tells you what each color or pattern means.
  • Circle graph (pie chart) — a circle cut into sections; each section shows a part of the whole. All sections add to 100\%.
  • Whole (total) — the full amount the graph describes, such as all 120 employees or the entire $3,200 paycheck.
  • Central angle — the angle at the center of the circle for one section. All central angles add to 360^\circ (e.g., half the circle = 180^\circ).
  • Percent — "out of 100." 35\% means \tfrac{35}{100} = 0.35.
  • Ratio — a comparison of two amounts, such as 48 cars to 12 bikes, or 4 : 1.

All Math words in the glossary

Reading a bar graph

A bar graph has two parts you must check before you look at any bar:

  1. The title and labels — What is being counted? Employees? Dollars? Thousands of dollars?
  2. The scale — What does each gridline stand for? Count the jump between labeled lines.

Reading a bar. Move your eyes straight across from the top of the bar to the scale. If the top lands exactly on a gridline, read that number. If it lands between two gridlines, estimate: halfway between 45 and 50 is 47.5; a bit below 50 might be 48.

Example: In the commute graph below, the scale counts by 5s. The Train bar ends exactly on 30. The Car bar ends a little below 50, at 48.

Bar graph titled How 120 employees get to work. Vertical axis: number of employees, 0 to 50 in steps of 5. Bars, each labeled with its value: Car 48, Bus 22, Train 30, Bike 12, Walk 8.05101520253035404550Car48Bus22Train30Bike12Walk8Number of employeesHow employees get to work

Look at the figure: slide straight across from the top of each bar to the scale.

Reading bars on the commute bar graph. The scale counts by 5s. A dashed line goes straight across from the top of the Train bar to exactly 30. Another dashed line from the top of the Car bar lands a little below the 50 gridline, between 45 and 50, so the Car bar reads 48.05101520253035404550CarBusTrainBikeWalkNumber of employeesHow employees get to work4830(a bit below 50)

Comparing bars. Most questions ask you to combine two or more bars:

Question wordsOperationExample (commute graph)
how many more / fewer, differencesubtractCar − Bus = 48 - 22 = 26
total, altogether, combinedaddBike + Walk = 12 + 8 = 20
how many times as manydivideCar ÷ Bike = 48 \div 12 = 4
what fraction / percent of allpart ÷ total\tfrac{30}{120} = 25\% take the train

Finding the total. Many questions need the total. Add every bar: 48 + 22 + 30 + 12 + 8 = 120 employees.

Double bar graphs. Each category has two bars, one for each group. Use the legend to know which is which. To compare the groups in one category, subtract the two bars in that category. The biggest difference is the biggest gap between a pair of bars — it is not always where the tallest bar is.

Watch the units. If the axis says "Sales (thousands of $)," a bar at 70 means 70 \times 1{,}000 = \$70{,}000. Read the axis title every time.

Reading a circle graph

A circle graph shows how a whole is split into parts. The whole circle is 100\% of something — a budget, a group of people, a day of time. Each section is labeled with a percent (sometimes with a fraction or an amount).

Circle graph of a family's monthly budget: Rent 35%, Food 15%, Transportation 12%, Utilities 8%, Savings 10%, Other 20%.Monthly BudgetRent 35%Food 15%Transportation 12%Utilities 8%Savings 10%Other 20%

Rule 1 — The percents add to 100%. In the budget graph: 35 + 15 + 12 + 8 + 10 + 20 = 100. If one section is missing its label, subtract the others from 100\%.

Example: A graph shows 45\%, 30\%, and 10\% with one unlabeled section. The missing section is 100 - 45 - 30 - 10 = 15\%.

Rule 2 — Section amount = percent × whole. Change the percent to a decimal by moving the decimal point two places left, then multiply.

Example: The family earns $3,200. Rent is 35\%: 0.35 \times 3{,}200 = \$1{,}120. Food is 15\%: 0.15 \times 3{,}200 = \$480.

Rule 3 — Compare sections with percents first. Two sections of the same circle share the same whole, so you may add or subtract the percents first. Rent is 35\% - 15\% = 20\% more than food, and 0.20 \times 3{,}200 = \$640.

Rule 4 — Work backwards: whole = section amount ÷ percent. If you know the amount of one section and its percent, you can find the whole.

Example: 36 people chose fish, which is 15\%. The whole is 36 \div 0.15 = 240 people.

Rule 5 — You cannot compare two different circles by percent alone. 20\% of a $5,000 budget ($1,000) is more money than 30\% of a $2,000 budget ($600). A bigger percent does not mean a bigger amount unless the wholes are equal.

Central angles and changing a bar graph into a circle graph

A circle has 360^\circ all the way around. Each section of a circle graph gets the same share of the 360^\circ as it has of the data.

Central angle = (section's share as a decimal) × 360°

Example: A section of 20\% has angle 0.20 \times 360^\circ = 72^\circ. A section of 25\% (one quarter) has 90^\circ — a right angle. A section of 50\% has 180^\circ — a straight line.

Three circles showing central angles. A 20% section has a 72 degree angle. A 25% section, one quarter, has a 90 degree right angle marked with a small square. A 50% section has a 180 degree angle, a straight line.72°20% × 360° = 72°90°25% × 360° = 90°180°50% × 360° = 180°

Going the other way: If a section's angle is 54^\circ, its share is \tfrac{54}{360} = 0.15 = 15\%.

Turning a bar graph into a circle graph takes three steps:

  1. Total all the bars.
  2. Divide each bar by the total to get its share.
  3. Multiply each share by 360^\circ.

Example: Store A sold 40, 55, 70, and 35 (thousand dollars) in four quarters. Total = 200.

QuarterShare of totalCentral angle
Q140 \div 200 = 20\%0.20 \times 360 = 72^\circ
Q255 \div 200 = 27.5\%0.275 \times 360 = 99^\circ
Q370 \div 200 = 35\%0.35 \times 360 = 126^\circ
Q435 \div 200 = 17.5\%0.175 \times 360 = 63^\circ

Check: 72 + 99 + 126 + 63 = 360^\circ ✓. The shares add to 100\% and the angles add to 360^\circ — always use this as a self-check.

Circle graph of Store A's sales, total 200 thousand dollars. Q1 40 thousand is 20% with a 72 degree angle; Q2 55 thousand is 27.5% with 99 degrees; Q3 70 thousand is 35% with 126 degrees; Q4 35 thousand is 17.5% with 63 degrees. The angles add to 360 degrees.Q172°Q299°Q3126°Q463°Store A (total 200)Q1: 40 ÷ 200 = 20%Q2: 55 ÷ 200 = 27.5%Q3: 70 ÷ 200 = 35%Q4: 35 ÷ 200 = 17.5%72 + 99 + 126 + 63= 360° ✓

Which graph to use? A bar graph is best for comparing amounts between categories. A circle graph is best for showing parts of one whole. The GED may ask which display fits a situation: use a circle graph only when the parts add up to a meaningful whole.

Formula

Formulas to memorize

\text{Percent of whole} = \frac{\text{part}}{\text{whole}} \times 100\% \text{Section amount} = \text{percent (as a decimal)} \times \text{whole} \text{Whole} = \frac{\text{section amount}}{\text{percent (as a decimal)}} \text{Central angle} = \frac{\text{part}}{\text{whole}} \times 360^\circ
  • part — the amount in one category (one bar or one section)
  • whole — the total of all categories
  • 360° — the number of degrees in a full circle

None of these are on the GED formula sheet. They all come from one idea — a part divided by the whole — so learn that idea and the four versions follow.

Worked example

Example 1 (easy): Read and compare bars

Using the commute graph, how many employees either bike or walk to work?

  1. Bike bar = 12 (just above the 10 gridline, below 15).
  2. Walk bar = 8 (between 5 and 10).
  3. "Either … or" combines the groups, so add: 12 + 8 = 20.
Example 1. The Bike bar (12) and the Walk bar (8) are stacked into one bar of 20 employees and placed beside the Bus bar (22). The stacked bar is a little shorter than the Bus bar.0510152025Number of employees12Bike8WalkBike 12Walk 8Bike + Walk= 2022Bus

Answer: 20 employees.

Check: 20 is less than the Bus bar (22), and the two short bars together look about as tall as the Bus bar ✓

Worked example

Example 2 (medium): Percent from a circle graph — real world

The Ramirez family's take-home pay is $3,200 a month (see the budget graph). How much do they spend on transportation and utilities combined?

  1. Add the two percents: 12\% + 8\% = 20\%.
  2. Change to a decimal: 20\% = 0.20.
  3. Multiply by the whole: 0.20 \times 3{,}200 = 640.
Example 2. The Ramirez family's budget circle graph with a whole of 3,200 dollars. The Transportation (12%) and Utilities (8%) sections are shaded together: 12% + 8% = 20%, and 0.20 × 3,200 = 640 dollars.Rent35%Food15%Transp.12%Util.8%Savings10%Other20%Whole = $3,200Transportation 12%+ Utilities 8%= 20%0.20 × 3,200= $640

Answer: $640 per month.

Check: Separately: 0.12 \times 3{,}200 = 384 and 0.08 \times 3{,}200 = 256; 384 + 256 = 640 ✓

Worked example

Example 3 (medium): Double bar graph

Double bar graph of quarterly sales in thousands of dollars. Store A: Q1 40, Q2 55, Q3 70, Q4 35. Store B: Q1 30, Q2 60, Q3 45, Q4 65. Vertical axis 0 to 80 in steps of 10.01020304050607080Q1Q2Q3Q4Store AStore BSales (thousands of $)

What were the two stores' combined sales in Q2, in dollars?

  1. Q2: Store A = 55, Store B = 60 (thousands of dollars).
  2. Add: 55 + 60 = 115 thousand.
  3. Convert using the axis label: 115 \times 1{,}000 = \$115{,}000.

Answer: $115,000.

Check: Both bars are a little above 50, so the sum should be a bit more than 100 thousand ✓

Worked example

Example 4 (GED-hard): Work backwards to the whole

A circle graph shows how a club spent its budget. The Food section is labeled 25\%, and the club spent $450 on food. The Prizes section has a central angle of 54^\circ. How much did the club spend on prizes?

  1. Find the whole budget from the food section: \text{whole} = 450 \div 0.25 = 1{,}800.
  2. Change the prizes angle to a share: \dfrac{54}{360} = 0.15.
  3. Find the prizes amount: 0.15 \times 1{,}800 = 270.
Example 4. A club's budget circle graph. The Food section is 25% (a 90 degree right angle) and equals 450 dollars, so the whole budget is 450 ÷ 0.25 = 1,800 dollars. The Prizes section has a 54 degree central angle, which is 54/360 = 15% of the whole, or 270 dollars. The rest of the circle is other spending.54°Food 25%$450PrizesOther spendingWhole = 450 ÷ 0.25= $1,800Prizes share = 54 ÷ 360= 0.150.15 × 1,800= $270

Answer: $270 on prizes.

Check: 270 \div 1{,}800 = 0.15, and 0.15 \times 360^\circ = 54^\circ ✓

Worked example

Example 5 (GED-hard): Bar graph to circle graph

Using the commute graph (total 120 employees), what central angle would the Bus section have in a circle graph?

  1. Share of employees who ride the bus: \dfrac{22}{120}.
  2. Multiply by 360^\circ: \dfrac{22}{120} \times 360^\circ = 22 \times 3 = 66^\circ (because 360 \div 120 = 3).
Example 5. The commute data (120 employees) drawn as a circle graph. Each employee is worth 360 ÷ 120 = 3 degrees. Car 48 × 3 = 144 degrees, Bus 22 × 3 = 66 degrees (highlighted), Train 30 × 3 = 90 degrees, Bike 12 × 3 = 36 degrees, Walk 8 × 3 = 24 degrees.Car144°Bus66°Train90°Bike 36°Walk 24°360° ÷ 120 = 3°per employeeBus: 22 × 3°= 66°All: 120 × 3° = 360° ✓

Answer: 66°.

Check: Each employee is worth 3^\circ. All employees: 120 \times 3 = 360^\circ ✓ Bus: 22 \times 3 = 66^\circ ✓

Calculator tip

Using the TI-30XS MultiView

Most bar-graph reading is done by eye, and the arithmetic is small enough to do on paper. The calculator helps with percents and angles:

  • Percent of a whole: type the percent as a decimal. For 35\% of $3,200 press 0.35 × 3200 enter → 1120.
  • Central angle: for \tfrac{22}{120} of 360^\circ, use the fraction key: n/d 22 ↓ 120 → × 360 enter → 66. Or simply type 22 ÷ 120 × 360 enter.
  • Work backwards: for 36 \div 15\%, type 36 ÷ 0.15 enter → 240.
  • If an answer shows as a fraction, press toggle to switch to a decimal.

When not to use it: Do not use the calculator to "read" the graph — the most common errors are reading the wrong bar or the wrong scale, and no calculator can fix that. Write down the values you read first, then calculate.

Common mistake

Common mistakes

  • Reading the wrong bar. In a busy graph it is easy to slide to the neighboring bar. Wrong: Car − Bus when the question says Car − Train. Right: put your finger (or cursor) on each category label before reading.
  • Treating a count as a percent. Wrong: "8 people walk, so 8% walk." Right: percent = 8 \div 120 \approx 6.7\%. A count equals a percent only when the total is exactly 100.
  • Ignoring the units on the axis. Wrong: "Store A sold $70 in Q3." Right: the axis says thousands, so it is $70,000.
  • Using 100 instead of 360 for an angle. Wrong: a 20\% section has a 20^\circ angle. Right: 0.20 \times 360^\circ = 72^\circ.
  • Forgetting a section when finding a missing percent. Wrong: 100 - 45 - 30 = 25\% (skipped the 10% section). Right: 100 - 45 - 30 - 10 = 15\%.
  • Confusing the tallest bar with the biggest difference. The biggest gap in a double bar graph is where the two bars are farthest apart, not where one bar is tallest.
Test strategy

GED test strategy

  • Read the title, axis labels, legend, and scale first — before you read the question's numbers. Ten seconds here prevents most mistakes.
  • Read the question twice. Underline the words "more," "total," "percent," "of all." They tell you the operation.
  • Write the values down. Jot a mini-table of the bars or sections you need. Then do the math from your table, not from the picture.
  • Estimate to eliminate. If Rent is a bit more than one-third of $3,200, the answer must be a bit more than $1,067. Cross out choices that are far off.
  • Use the 100% / 360° checks. Percents of a circle graph add to 100; angles add to 360. If your answers don't, you made an error.
  • Plug answers back in for work-backwards questions: if the answer choice is 240 people, does 15\% of 240 give 36?
  • Watch your time. Graph questions can be quick; don't spend more than two minutes on one. Flag it and come back.

Check yourself

Bar graphs and circle graphs

The bar graph shows how the 120 employees of a company get to work.

Bar graph titled How 120 employees get to work. Vertical axis: number of employees, 0 to 50 in steps of 5. Bars, each labeled with its value: Car 48, Bus 22, Train 30, Bike 12, Walk 8.05101520253035404550Car48Bus22Train30Bike12Walk8Number of employeesHow employees get to work

How many more employees drive a car than take the train?

Answer choices
Hint

Read the top of each bar against the gridlines on the left. Then decide which operation "how many more" calls for.

Check yourself

Bar graphs and circle graphsFill in

The bar graph shows how the 120 employees of a company get to work.

Bar graph titled How 120 employees get to work. Vertical axis: number of employees, 0 to 50 in steps of 5. Bars, each labeled with its value: Car 48, Bus 22, Train 30, Bike 12, Walk 8.05101520253035404550Car48Bus22Train30Bike12Walk8Number of employeesHow employees get to work

What percent of the employees drive a car to work?

%

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Percent = part ÷ whole × 100. The whole is all the employees, not just the tallest bar.

Summary

  • Check the title, labels, scale, and legend before reading any bar or section.
  • Bar graphs show amounts: subtract for "how many more," add for "total," divide by the total for "what percent."
  • Circle graphs show parts of one whole: section amount = percent \times whole; the percents add to 100\%.
  • Work backwards with whole = amount \div percent.
  • Central angle = \dfrac{\text{part}}{\text{whole}} \times 360^\circ; all angles add to 360^\circ.
  • Watch units like "thousands of dollars" and never confuse a count with a percent.
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