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

If you need further help, just post back with the areas of the circles and I'll post the calculations you need. In summary, the conversation discusses a problem with finding the area of three intersecting circles and the steps needed to solve it. The solution involves calculating the areas of each individual circle and their intersections, and then using a formula to find the final area.
  • #1
airborne18
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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

 
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  • #2
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]
Circle B: [itex]s^2 = (x-c)^2 + (y-d)^2[/tex]
Circle C: [itex]t^2 = (x-e)^2 + (y-f)^2[/tex]

Let A be the area of circle A, B be the area of circle B, and C be the area of circle C.

Call the intersection between circles A and B, area D (the football-like shape)
Similarly, call the intersection between circles B and C, area E and the intersection between circles A and C, area F.

Lastly, call the intersection of all 3 circles (the diamond-like shaped area in the center), area G.

If I understand your question correctly, you are looking for A + B + C - D - E - F + 2G

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.

At this point, I'll leave the math to you.
 

Attachments

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1. What is the formula for finding the area of intersecting circles?

The formula for finding the area of intersecting circles is A = πr2 - ½d1d2, where r is the radius of each circle and d1 and d2 are the distances between the centers of the circles.

2. How do you determine the intersection points between two circles?

The intersection points between two circles can be determined by using the Pythagorean theorem to find the length of the line segment connecting the centers of the circles. If this length is equal to the sum of the radii of the circles, then the circles intersect at two points.

3. Can the area of intersecting circles be negative?

No, the area of intersecting circles cannot be negative. If the formula for finding the area yields a negative value, it means that the circles do not intersect and the area is equal to 0.

4. How does the location of the intersection points affect the area of intersecting circles?

The location of the intersection points can affect the area of intersecting circles. If the intersection points are closer to the center of the circles, the area of the intersection will be larger. If the intersection points are closer to the edges of the circles, the area of the intersection will be smaller.

5. Can the area of intersecting circles be greater than the combined area of the individual circles?

Yes, the area of intersecting circles can be greater than the combined area of the individual circles. This is because the formula for finding the area of intersecting circles takes into account the overlapping area of the circles, which can be greater than the non-overlapping areas of the individual circles.

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