Fun Problem: Area of Overlapping circles

In summary, the area of overlapping circles can be calculated by finding the area of each individual circle and subtracting the area of the intersecting region using the formula A = πr^2. The formula for finding the area of intersecting circles is A = 2r^2 * (cos^-1(d/2r) - (d/2r) * √(1 - (d/2r)^2)) and it cannot be negative. There is no limit to the number of circles that can overlap and this concept is used in various fields such as science, engineering, and leisure activities.
  • #1
relativitydude
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Picture two indentical circles with their radii overlapping. They form an intersection, what is the area of their intersection?

I solved it the calculus route and it can be solved geometrically. Have fun.
 
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  • #2
Use the cosine rule to find the two subtended angles. From these you can find the areas of the sectors as well as the triagles subtended at the centers. Subtracting, gives the ares of the two caps (segments?). Add these areas to find the area of intersection.
 
  • #3
Please elaborate?
 

FAQ: Fun Problem: Area of Overlapping circles

1. How do you calculate the area of overlapping circles?

The area of overlapping circles can be calculated by finding the area of each individual circle and then subtracting the area of the intersecting region. This can be done using the formula A = πr^2, where r is the radius of the circle.

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

The formula for finding the area of intersecting circles is A = 2r^2 * (cos^-1(d/2r) - (d/2r) * √(1 - (d/2r)^2)), where r is the radius of the circles and d is the distance between their centers.

3. Can the area of overlapping circles be negative?

No, the area of overlapping circles cannot be negative. It is always a positive value representing the amount of space covered by the overlapping region.

4. Is there a limit to the number of circles that can overlap?

No, there is no limit to the number of circles that can overlap. The area of overlapping circles can be calculated for any number of circles as long as their centers are within the radius of each other.

5. How is the area of overlapping circles used in real life?

The concept of the area of overlapping circles is used in many areas of science and engineering, such as in optics for calculating the intensity of light, in biology for studying cell growth and division, and in architecture for designing complex structures. It is also commonly used in recreational activities, such as creating Venn diagrams and finding the area of a pool cover.

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