Overlapping Circles
Lune One-Half the Area of the Circle
Given two circles, each with radius r, locate the center of one on a radius of the other so that lunes are formed and each of the lunes has an area one-half the area of the circle.
That is, how far apart are the centers of the circles when the overlap amounts to one-half of the area of a circle?
The image to the right shows the approximate placement.
Is this configuration constructible with straightedge and compass construction?
Hint: Construct segment AC in the image. The segment of the circle cut off by the segment AC is one-fourth of the area of the circle.
Construct a GSP file to implement your construction.
The formula for the area of the SEGMENT of a circle with central angle
For our problem with the area of the lune being one-half of the area of the circle, there is no loss of generality in letting the radius r = 1.
A. This is an excellent opportunity to use a spreadsheet for finding an approximate size of the central angle.
B. Even better, it is an easy example to implement the use of a spreadsheet to do an iteration of the calculation of the size of the central angle.
NOTE, HOWEVER, THAT THE PROBLEM ASKS FOR HOW FAR APART THE TWO CENTERS WILL BE. CAN THIS BE READILY DETERMINED ONCE YOU KNOW THE SIZE OF THE CENTRAL ANGLE?
C. As the center of the second circle moves along the radius, in a certain range, the area of the overlap increases from 0 to the area of the circle. Express this increase as a function in the range of the central angle from 0 to π. Click HERE to see a graph of this function.
