If you remember how Venn Diagrams looked with the three intersecting circles, that is the context for this puzzle.

There are three circles, circle A, circle B, and circle C. Each circle passes through the center of the other two. What is the area of the intersection of these three circles?

The figure that forms is an equilateral triangle with arc segments from vertex to vertex of radius equal to the triangle´s side. First we obtain the area of the equilateral triangle (A1), then we use an area-angle linear relation on one of the circles, that is, if an equilateral triangle has its tree angles equal to 60 degrees, then the area of a circle segment with an aperture of 60 degrees is the sixth part of the circle´s area (A2). If we subtract A2-A1 we get a third area of a circle´s arc section (A3), adding 3 times A3 with A1 we get the final area of the figure formed by the intersecting circles.
A1=√3*R²/4, A2=Pi*R²/6, A3=A2-A1=R²*(2*Pi-3*√3)/12
Total area = 3*A3+A1=R²*(Pi-√3)/2

Posted by Antonio on 2003-09-03 07:41:33