## Area of overlap of circles of equal radius where each touches the centre of the other

Problem: Given 2 circles that intersect each other in such a way that each circle passes through the other’s centre. What is the area of overlap if the radius of each is 4cm? Solution:

If we draw a line segment between the points of intersection of the circles it divides the required area in half. The portion on the left of this segment is a sector of the circle on the right and the area on the right is a sector of the circle on the left. Both sectors have the same area.

We can build Equilateral triangles with their vertices at the circle centres and the points of intersection, as shown below, whose side length is r: The angle <AC1B is therefore 120°.

The red colour below outlines a sector of circle C1 with an angle of 120 degrees. The portion of this on the right of [AB] is a segment of the circle on the left. This is half of the required area.

The area of the segment is equal to the area of the sector minus the area of triangle AC1B.

= π r2 (120/360°) – (r/2 x (r√3)/2)

= π r2/3 – (r2√3) /4

= r2( π /3 – √3 /4)

The total required area is twice this. = 2r2( π /3 – √3 /4)

Finally sub in the value for the radius of 4cm and let the calculator do the rest.

=17.197 cm2

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