3 Circle Venn Diagram: Regions, Examples and Template

A 3 circle Venn diagram is three overlapping circles, each one holding a set of things. The overlaps show what the sets share. It is the classic picture for sorting items into groups, and it is a favorite in math, science and language lessons. This guide names every region, walks through two worked examples with the numbers, shows how to answer the usual questions, and gives you a free template to start from.

What a 3 Circle Venn Diagram Shows

A Venn diagram, also called a set diagram or logic diagram, shows all possible logical relations between a finite collection of sets. That is how Wikipedia defines it. The sets are drawn as overlapping closed curves, usually circles. A point inside a curve is an element of that set, and a point outside the curve is not.

John Venn conceived these diagrams around 1880 and popularized them in his book Symbolic Logic in 1881, according to Wikipedia. Similar ideas came earlier from Christian Weise in 1712 and Leonhard Euler in 1768. Today Venn diagrams are used to teach elementary set theory and to show set relationships in probability, logic, statistics, linguistics and computer science.

The Three Sets

You pick three groups to compare. We call them A, B and C here, but in real use they get real names: Chess, Robotics and Choir, or Even, Multiple of 3 and Prime. Each circle is one group, and every item you list goes in the circle or circles it belongs to.

Why There Are Seven Regions Plus the Outside

A Venn diagram for n sets must contain every possible combination of being in or out of each set. That is 2 to the power of n zones, Wikipedia notes. For three sets it is 2 x 2 x 2 = 8 zones. Seven of them are inside the circles. The eighth is the space outside all three circles, for items that belong to none of the sets.

That is the reason the three circles are placed the way they are. Every pair overlaps, and all three overlap in the middle, so no combination is missing.

The Seven Regions, Named

Here is every region of a 3 circle Venn diagram. The words “only” and “and” do the work, so read them carefully.

Two habits will save you from most mistakes. First, the phrase “A and B” on its own usually includes the middle region, because an item in all three sets is also in A and in B. Second, when a question says “only”, exclude the middle.

Worked Example: Sort Numbers Into Three Sets

Start with the numbers 1 to 9 and three sets: even numbers, multiples of 3 (called Times 3 in the diagram), and prime numbers. Here is the finished diagram, drawn from the data below it.

Work out where each number goes by asking three yes or no questions.

1. 2 is even and prime, but not a multiple of 3. It goes in the overlap of Even and Prime.
2. 3 is a multiple of 3 and prime, but not even. It goes in the overlap of Multiple of 3 and Prime.
3. 6 is even and a multiple of 3, but not prime. It goes in the overlap of Even and Multiple of 3.
4. 4 and 8 are only even.
5. 9 is only a multiple of 3.
6. 5 and 7 are only prime.
7. 1 is in no set. It is outside the circles.

No number is even, a multiple of 3 and prime at once, so the middle region is empty. That is a real result, not a mistake. An empty region tells you something: nothing in this data belongs to all three sets.

Open this 3 circle Venn diagram in the Venn diagram maker

Worked Example: Three School Clubs

Here is a second example with real names, where the middle region is not empty. Twelve students may join the Chess, Robotics and Choir clubs. One student, Lee, joins none, so Lee is outside the circles.

Read it region by region:

• Chess only: Ana and Ben.
• Robotics only: Gus and Hana.
• Choir only: Ivy and Jo.
• Chess and Robotics only: Cara.
• Chess and Choir only: Dev.
• Robotics and Choir only: Eli.
• All three: Fay and Kim.

In the diagram above, the tool sorted each name by itself. You type a name once in each column where it belongs, and the maker puts it in the right region.

Show Counts Instead of Names

Names get crowded when a set has dozens of members. Switch the display to counts and each region shows how many items it holds instead.

The totals here are 6 for each club. Check one: Chess has Ana and Ben (2), plus Cara (1), plus Dev (1), plus Fay and Kim (2), and 2 + 1 + 1 + 2 = 6.

Open the club counts in the Venn diagram maker

How to Answer a 3 Circle Venn Diagram Question

Test questions usually give you numbers and ask for a missing region, or give you a picture and ask you to count. The method is the same either way. Work from the middle outward.

1. Fill the middle first. The count for “all three” is the most specific piece of information, so it goes in first.
2. Fill the three overlaps of two circles. If you are told that 8 students are in Chess and Robotics, that number includes the students who are in all three. Subtract the middle to get the “only” count. In the clubs example, 3 students are in Chess and Robotics, and 2 of them are in all three, so 1 is in Chess and Robotics only.
3. Fill the circles’ own parts. Take the size of the whole set and subtract everything already placed inside that circle.
4. Put the rest outside. Anything in none of the sets goes outside the circles.
5. Check. Add every region, the outside included. The total must equal the number of items you started with.

The most common mistake is to write the total of a pair straight into its overlap without subtracting the middle. It counts the center twice, and the numbers stop adding up.

The Formula for Three Sets

When you know the set sizes and the overlaps, you can count everything inside the circles without drawing anything. The rule is called the inclusion-exclusion principle. For three sets it says:

Total in A, B or C = A + B + C, minus (A and B), minus (A and C), minus (B and C), plus (A and B and C).

Why does it work? Add A, B and C, and everything in two circles is counted twice, while the middle is counted three times. Subtract each pair overlap and the middle is now subtracted too often, so it is added back once. Wikipedia describes exactly this correction for the three-set case.

Try it on the numbers example. Even has 4 numbers (2, 4, 6, 8), Multiple of 3 has 3 (3, 6, 9), and Prime has 4 (2, 3, 5, 7), so A + B + C = 11. Even and Multiple of 3 share 1 number (6). Even and Prime share 1 (the number 2). Multiple of 3 and Prime share 1 (the number 3). All three share 0. So 11 minus 1 minus 1 minus 1 plus 0 = 8. And 8 is right: the numbers 1 to 9 with the number 1 taken out.

Try it on the clubs. The three clubs have 6 students each, so A + B + C = 18. Each pair of clubs shares 3 students, so subtract 9. All three share 2, so add 2. That gives 18 minus 9 plus 2 = 11, which matches the 11 students inside the circles.

A Region Can Be Empty

Beginners often think an empty region means the diagram is wrong. It does not. The factors of 12, 18 and 30 make a good example. Every number that divides two of them also divides the third, so the regions for exactly two circles have nothing in them.

The middle holds 4 numbers: 1, 2, 3 and 6. Two numbers, 4 and 12, divide only 12. Two, 9 and 18, divide only 18. Four, 5, 10, 15 and 30, divide only 30. An empty region is the answer to a question such as “which numbers divide 12 and 18 but not 30?” There are none.

A Free 3 Circle Venn Diagram Template

You can start from a blank diagram. This link opens the Venn diagram maker with three sets named Set A, Set B and Set C and six placeholder items already sorted into different regions. Rename the sets, type your own items, and the picture updates as you type.

Open the free 3 circle Venn diagram template

What the tool does:

• Draws two or three circles. Pick 3 for this diagram.
• Sorts items into regions for you: type an item in every column where it belongs.
• Shows the items or only the counts, with optional totals for each set.
• Lets you change the colors, the transparency of the circles and the canvas size.
• Downloads a PNG or an SVG, copies the image, or copies a link to your exact diagram.
• Runs in your browser with no sign-up.

One limit to know about: the circles are the same size whatever a set holds. An area-proportional (scaled) Venn diagram draws each shape’s area in proportion to its number of elements. Ours does not. Wikipedia notes that ordinary Venn diagrams are schematic and generally not drawn to scale, so read the numbers, not the areas.

4 Circle and 5 Circle Venn Diagrams

People search for four-circle and five-circle diagrams, so here is what we can say from the sources we read. Wikipedia says Venn diagrams typically represent two or three sets, with forms that allow more. It shows a four-set arrangement of circles that has only 14 regions instead of the 16 a four-set Venn diagram needs, because no region has only the yellow and blue circles, or only the red and green. That arrangement counts as an Euler diagram, not a Venn diagram. Venn himself designed a four-set diagram with ellipses.

So do not expect four plain circles to give you every combination. We did not find a source that proves no arrangement of four circles can work, so we say only what Wikipedia shows. Our own maker draws two or three circles, and we do not offer four or five sets.

Venn Diagram or Euler Diagram?

The two are close cousins. In a Venn diagram the curves overlap in every possible way. Diagrams that do not show every relation are Euler diagrams, so a Venn diagram is a special case of an Euler diagram. In practice, if you draw three circles that always overlap and then write “nothing” or leave a region empty, you have a Venn diagram. If you redraw the circles so that unrelated sets do not touch, you have an Euler diagram.

Where 3 Circle Venn Diagrams Are Used

• Math class. Sorting numbers by properties, as in the first example, and practicing set notation.
• Statistics and probability. Showing events that can overlap, such as students who take math, science or both.
• Language lessons. Comparing three texts, characters or word families.
• Planning and research. Seeing which customers use which of three products, or which skills a team shares.
• Logic and computer science. Showing what a search or a database query with AND and OR includes.

Venn diagrams are one of the chart types in our overview of types of graphs and charts, where you can see how they compare with bar charts, pie charts and timelines.

Common Mistakes

• Counting the middle three times. When you add three set sizes, the center is counted three times. Use the formula above.
• Writing “A and B” when you mean “A and B only”. Say “only” and exclude the middle when you mean it.
• Forgetting the outside. Items in no set still count toward the total.
• Reading area as size. Circles are the same size whatever the count, so compare the numbers.
• Too many words in a region. A region with a long list is hard to read. Switch to counts or shorten the labels.
• Circles that do not overlap in every way. If a pair does not overlap, the diagram is an Euler diagram, and the reader may miss that a combination is possible.

Summary

A 3 circle Venn diagram has three sets, seven regions inside the circles and one outside. To use it, name the regions, fill the middle first, work outward and check that the totals add up. For counts, use the inclusion-exclusion formula. When you are ready to draw your own, the Venn diagram maker sorts your items for you and exports a PNG or SVG.