The crescent area created by overlapping circles

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SUMMARY

The area of the crescent created by overlapping circles can be calculated using geometric principles. For two circles with equal radius, such as 50µm, and a center offset of 10µm, the area can be determined in two steps. First, calculate the area bounded by the arc and the chord formed by the intersection points. Second, sum the areas from both circles to find the total overlapping area. This method provides a precise approach to determining the crescent area.

PREREQUISITES
  • Understanding of basic geometry concepts, including chords and arcs.
  • Familiarity with circle properties and area calculations.
  • Knowledge of geometric formulas for area and triangle calculations.
  • Ability to visualize and interpret overlapping shapes in a two-dimensional space.
NEXT STEPS
  • Research geometric formulas for calculating the area of a circle segment.
  • Explore methods for finding intersection points of two circles.
  • Learn about advanced geometric concepts such as the Law of Cosines for triangle area calculations.
  • Investigate software tools for visualizing geometric shapes and their properties.
USEFUL FOR

Mathematicians, geometry enthusiasts, and students studying advanced geometry concepts will benefit from this discussion, particularly those interested in calculating areas of overlapping shapes.

SimonHollas
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Dear All,

I need to know the area of the crescent created by overlapping circles;e.g. a circle radius 50µm overlapped by an equal circle with its centre 10µm to the left.
Any help you can offer would be gratefully received,

thanks.
 
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Two steps:

1. Consider a circle with center O, and a chord AB (ie, a line segment between two points on the circle). Find the area bounded by the arc AB and the line AB, ie, the area inside the circle and on one side of the chord. This can be done by finding the area of the pie slic corresponding to the arc AB and subtracting the area of the triangle ABO.

2. Given two circles, the two points where they intersect form a chord on both circles, and sum of the two corresponding areas from 1 gives the area of overlap.
 

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