Knowing all areas of three overlapping circles excepting the area of $A'∩B'∩C$ and the area of $C$, is it possible to find the area of $A'∩B'∩C$? Let's imagine I have three overlapping circles. I know the following information: $1.$ the area of circle $A$ = the area of circle $B = 1$ $2.$ the area of $A\cap B =\frac 15$ $3.$ the area of $A\cap C =$ the area of $B\cap C = \frac 9{20}$ $4.$ the area of $A\cap B \cap C=x$ such that $0\lt x\lt \frac 15 $ $5.$ the area of circle $C$ is unknown. Is it possible, using this information, to find the possible range of areas for $A'\cap B'\cap C$, and thus find the range of areas for $C$? **EDIT:** I am aware of some methods whereby you can measure the overlap of three circles. I'm beginning to wonder if there's a way to do it in reverse to find $C$.

There is no upper limit for the area of $C$. The center of $C$ lies on the line $r$ passing through the intersection points of circumferences $A$ and $B$, and the arc of $C$ inside $A$ approaches a straight line when the radius of $C$ tends to infinity. That limiting line is perpendicular to $r$ and cuts $A$ so that the lesser part is 45% of $A$. This situation is compatible with your constraints because $\hbox{area}(A\cap B\cap C)=0.45\ \hbox{area}(A\cap B)=0.09$.

A lower limit can be found, because it corresponds to the situation when the three circumferences have a point in common. Again, the bound on the area of $A\cap B\cap C$ is unimportant, but the area of $A\cap C$ is $0.45$ and this implies the radius of $C$ cannot be too small. Experimentally the lowest area turns out to be around $0.72$.