Can you draw 3 overlapping circles (Venn diagram) such that all 7 of the formed regions have the same area?

Venn diagram

For the case of two circles and 3 equal regions I found this answer:


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    $\begingroup$ My intuition tells me that this isn't possible, but I dunno how to prove it this late at night. Great concept, regardless! $\endgroup$
    – Brandon_J
    Commented Sep 18, 2019 at 5:46
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    $\begingroup$ Well, if you remove the restriction that the 3 areas be circles, it's certainly possible. There's no requirement that a Venn diagram be made only of circles, that's just the easiest and classic example. $\endgroup$ Commented Sep 18, 2019 at 13:57
  • $\begingroup$ @DarrelHoffman While there may be thousands of solutions, it might still be interesting to find a shape that can do that. $\endgroup$
    – Strawberry
    Commented Sep 18, 2019 at 16:34
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    $\begingroup$ @Strawberry (Irregular) pentagons will do this easily, for example. E.g. let all intersected regions (for 2 or all 3 areas) be equilateral triangles with side 1 (and height $\frac{\sqrt3}{2}$), and the remaining areas (belonging to only 1 area) be (isosceles) triangles with base 2 and height $\frac{\sqrt3}{4}$. $\endgroup$
    – trolley813
    Commented Sep 18, 2019 at 18:34
  • $\begingroup$ @trolley813 Oh, I see what you mean now. $\endgroup$
    – Strawberry
    Commented Sep 19, 2019 at 10:34

1 Answer 1


Well, that's impossible.


Let $A$ be an area of any of the 7 regions. Now, let us consider only 2 of the overlapping circles (removing 3rd circle for a while) and notice that their overlapping area is $2A$, and each of the non-overlapping parts (belonging to one of the circles, but not both) should also have an area of $2A$. So, the only way to arrange this circles (it will be true for any pair of the circles) is as in the linked question (the distance between their centers must be $2x$, where $x\approx0.403972$). So, the centers of the circles must form a right triangle with side $2x$. Plotting this gives the following graph:
enter image description here
Now it's clear that the regions have unequal area, even without any calculations (for example, that's because the bottom side of the grey curved triangle, where all 3 circles overlap, lies well below the x-axis, but the intersection points of red and green circles have the same y-coordinates, since their centers lie on the x-axis, so the grey area must be definitely greater then the brown one).
Python code for plotting: Try it online! (unfortunately it will not run there, since tio.run does not support external packages, like matplotlib).

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    $\begingroup$ Great work! I wasn't sure if this had a solution. $\endgroup$ Commented Sep 18, 2019 at 6:32
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    $\begingroup$ @DmitryKamenetsky Thanks! I've added a link to Python code used for plotting. $\endgroup$
    – trolley813
    Commented Sep 18, 2019 at 6:39
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    $\begingroup$ Got to be honest @DmitryKamenetsky - when I see the OP say "I wasn't sure if this had a solution" it makes me very wary about attempting to solve their future puzzles... $\endgroup$
    – Stiv
    Commented Sep 18, 2019 at 6:59
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    $\begingroup$ @Stiv Hmmm... ok I will remember this for my future puzzles and try to make sure they have a solution. For me proving that a puzzle doesn't have a solution is also just as interesting. $\endgroup$ Commented Sep 18, 2019 at 7:33
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    $\begingroup$ @DmitryKamenetsky I totally agree with you - as a researcher myself a negative finding is often just as useful as (if not more important than) a positive result :) However, in my opinion being 'not sure' if there is a solution to a puzzle is different to already knowing that there isn't one. When that's the case it seems to become more of an exercise in mathematical proof and I wonder whether maybe this type of question would be more appropiate on the Math SE rather than Puzzling... Just my opinion! :) $\endgroup$
    – Stiv
    Commented Sep 18, 2019 at 8:05

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