SOLUTION: Two circles both with radius 4, the centre of each circle lies on the circumference of the other, what is the exact area which is common to both circles?

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Question 536015: Two circles both with radius 4, the centre of each circle lies on the circumference of the other, what is the exact area which is common to both circles?
Answer by htmentor(1343) About Me  (Show Source):
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Two circles both with radius 4, the centre of each circle lies on the circumference of the other, what is the exact area which is common to both circles?
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The area of a circular sector = r%5E2%2Atheta+%2F2 where theta is the sector angle and r is the radius
The sector consists of a circular segment plus a triangular portion
[think of an ice cream cone where the cone is the triangular portion and the segment is the ice cream]
The height of the triangle = 2 [half the width of the overlapping portion]
The base of the triangle = 2%2Asqrt%284%5E2+-+2%5E2%29
So the area of the triangle = 2%2Asqrt%2812%29+=+4%2Asqrt%283%29
The angle theta = 2%2Aarccos%282%2F4%29 = 2%2Api%2F3
The area of the segment = the area of the sector - the area of the triangle
Area(segment) = 16%2Api%2F3+-+4%2Asqrt%283%29
There are two segments which make up the overlapping area, so the area common to both cirles is
8%2A%284%2Api%2F3+-+sqrt%283%29%29