How to determine the area of intersecting circles?

In summary, to calculate the area of intersection of two overlapping circles in terms of their radii, you need to determine the end points of the chord connecting the points of intersection and calculate the angle made by this chord to the center of each circle. Then, the area of the overlap can be calculated as the difference between the area of the sector and the area of the isosceles triangle within the sector for each circle.
  • #1
elegysix
406
15
I've been wondering how to calculate the area of intersection of two overlapping circles in terms of their radii. There's two cases I'm interested in:

The easier case:
Suppose there are two circles of radius R and r (R > r). The center of the larger circle is at the origin, and the center of the other circle is at (x,y)=(R,0). How can I find the area of the overlap in terms of R and r? I can't think of any clear way to do this by hand.

The more complicated case:
Do the same except let the center of the smaller circle lie between (R-r,0) < (x,y) < (R+r,0)

I've attached a drawing to help show the problem.
Thanks for any insight. This isn't for school, so there's no rush or anything. Just curious.

circles.jpg
 
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  • #3
More complicated case is irrelevant. The distance between the centers (and the radii) are all that matter.

General approach: Get end points of chord which connects the points of intersection of the circles.

For each circle, calculate the angle made by connecting these end points to the center. Now the problem is reduced to a pair of problems of obtaining for each circle the area of the sector minus the area of the isosceles triangle within the sector.
 

1. How do you find the area of intersecting circles?

To find the area of intersecting circles, you will need to first determine the overlapping area between the two circles. This can be done by finding the distance between the two circle centers and using trigonometry to calculate the angle of the intersecting region. Then, you can use the formula A = r^2 * (θ - sinθ), where r is the radius of the circles and θ is the angle of intersection, to find the area of the overlapping region. Finally, add the areas of each circle and subtract the overlapping area to get the total area of the intersecting circles.

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

The formula for finding the area of intersecting circles is A = r^2 * (θ - sinθ), where A is the area, r is the radius of the circles, and θ is the angle of intersection.

3. Can I use the same formula to find the area of multiple intersecting circles?

Yes, you can use the same formula to find the area of multiple intersecting circles. However, you will need to find the overlapping region for each pair of circles and add them together to get the total area of all intersecting circles.

4. Is there a simpler way to find the area of intersecting circles?

There are different approaches to finding the area of intersecting circles, such as using calculus or dividing the intersecting region into simpler shapes. However, the formula A = r^2 * (θ - sinθ) is the most straightforward method for calculating the area.

5. Why is it important to find the area of intersecting circles?

Finding the area of intersecting circles is important in various fields such as engineering, architecture, and physics. It allows us to understand the amount of overlap between two or more circles, which can help in designing structures, analyzing data, and solving mathematical problems.

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