How do I find the area of intersecting circles in a Venn Diagram?

[airborne18]

Homework Statement

I don't know why but my brain is having one of its moments and I can't work through this. Not even on paper anymore.

Okay so I have 3 intersecting circles. Like a Venn Diagram. How do I find the area of all three minus the instersecting parts. I know how to do two intersecting, but I am trying to break this down into an algorithm to process it in my brain.

This is not homework, but I am posting it here. I just need some help with sequencing the steps to solve it, and then my brain might click with again. I am mentally stuck with the application so need a kick start.

Homework Equations

The Attempt at a Solution

 

[zgozvrm]

You didn't specify whether the circles all had the same radius or not, so I'll assume they don't.

By "3 intersecting circles. Like a Venn Diagram" I assume you mean that circle A partially intersects both circles B and C, circle B partially intersects both circles A and C, and circle C partially intersects both circles A and B so that there are 7 distinct areas bounded by the circles and their intersections (see attachment).

Let's assume that circle A has a radius of r and it's center at (a,b)
Also, circle B has radius s and center (c,d)
And, circle C has radius t and center (e,f)

The formulas for the 3 circles are then:
Circle A: [itex]r^2 = (x-a)^2 + (y-b)^2[/tex]<br /> Circle B: [itex]s^2 = (x-c)^2 + (y-d)^2[/tex]<br /> Circle C: [itex]t^2 = (x-e)^2 + (y-f)^2[/tex]<br /> <br /> Let A be the area of circle A, B be the area of circle B, and C be the area of circle C.<br /> <br /> Call the intersection between circles A and B, area D (the football-like shape)<br /> Similarly, call the intersection between circles B and C, area E and the intersection between circles A and C, area F.<br /> <br /> Lastly, call the intersection of all 3 circles (the diamond-like shaped area in the center), area G.<br /> <br /> If I understand your question correctly, you are looking for A + B + C - D - E - F + 2G<br /> <br /> Note that when you subtract D, for instance, you are already subtracting area G at the same time. So, you again subtract area G when you subtract areas D and F. Therefore, you must add G back in twice.<br /> <br /> At this point, I'll leave the math to you.[/itex][/itex][/itex]