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 centers of the circles (which are also points of intersection of two circles each) form an equilateral triangle. The area in question includes this triangle as well as adjacent areas bounded by arcs of circles.

We can also consider the area as three overlapping pie slices (sectors). Each pie slice is 1/6 of one of the full circle's, with area πr²/6. Three of these make πr²/2, but that counts the central equilateral triangle three times instead of one, so we have to subtract out twice the area of that triangle. The triangle's sides are r, and the altitude is ((√3)/2)r, so its area is r²(√3)/4, and twice its area is r²(√3)/2.

So the area sought is πr²/2 - r²(√3)/2. This is about .704770923 r².

Posted by Charlie on 2003-05-06 03:12:53