Diagram Desk

3 circle Venn diagram

Three sets, seven inside regions, and a club-membership example.

ABC
Three circles make seven inside regions.

Why three circles feel harder

A three circle Venn diagram (also written 3-circle Venn diagram, or Venn diagram with three circles) adds one set and doubles the bookkeeping. Two sets give three inside regions. Three sets give seven: only A, only B, only C, A with B but not C, A with C but not B, B with C but not A, and the center where all three meet.

Circles still work at three. Place two side by side and the third below, overlapping both. If your center triangle disappears, the third circle is too low.

Example: three after-school clubs

A is robotics (12 students), B is debate (9), C is yearbook (11). You are given the overlaps: robotics and debate only, 2; robotics and yearbook only, 3; debate and yearbook only, 1; all three clubs, 2. Only-robotics is 12 minus 2 minus 3 minus 2, so 5. Only-debate is 9 minus 2 minus 1 minus 2, so 4. Only-yearbook is 11 minus 3 minus 1 minus 2, so 5.

Add the seven regions: 5+4+5+2+3+1+2 = 22 students who are in at least one club. If the school has 30 students in the grade, 8 sit outside all three circles. Write 8 in the margin so the diagram accounts for everyone.

Phrases that mean this layout

Search phrases such as 'venn diagrams with 3 circles', 'venn diagram with 3 circles', and 'three circle venn diagram' all refer to this seven-region figure. The drawing does not change when the hyphen moves. What changes the drawing is a fourth set, which is a different template.

Steps

  1. Sketch the center first. Lightly mark the triple overlap so you know the three circles actually share one region.
  2. Label the seven pockets. Use tiny pencil names. Fill data only after every pocket has a name.
  3. Subtract from the outsides. Each set total includes the center and the pair overlaps. Subtract those before you write 'only A'.
  4. Check the sum. The seven regions plus the outside should equal the universe. If not, an item was counted twice.

Questions

How many regions are in a 3 circle Venn diagram?
Seven inside the circles, plus the outside. The center is the intersection of all three sets.
Can I still use circles for three sets?
Yes. Circles show every overlap of three sets. Switch to ellipses when you add a fourth set.
What goes in the very center?
Only items that belong to A and B and C. An item that misses even one set belongs in a pair lens or an outer crescent.

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