Fun Problem: Area of Overlapping circles

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SUMMARY

The area of intersection between two identical circles can be calculated using both calculus and geometric methods. The cosine rule is essential for determining the subtended angles at the centers of the circles. By calculating the areas of the sectors and the triangles formed, one can derive the areas of the segments. The final area of intersection is obtained by summing the areas of these segments.

PREREQUISITES
  • Understanding of the cosine rule in geometry
  • Knowledge of calculus for area calculations
  • Familiarity with geometric shapes and their properties
  • Ability to calculate areas of sectors and triangles
NEXT STEPS
  • Study the application of the cosine rule in circle geometry
  • Learn how to calculate the area of a sector in a circle
  • Explore methods for finding the area of triangles given two sides and the included angle
  • Investigate calculus techniques for finding areas of complex shapes
USEFUL FOR

Mathematicians, geometry enthusiasts, students studying calculus and geometry, and anyone interested in solving problems related to the area of overlapping circles.

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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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.
 
Please elaborate?
 

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