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Problem Statement

Along the diameter of a circle you can construct circles with equal radii that are tangent to each other. The outermost circles in the string of circles will be tangent to the large circle. (Tangent means that the circles touch each other but do not cross over each other, nor do they leave any gaps.)

How does the combined area of all of the shaded circles relate to the area of the entire circle?

Problem setup

What is the relationship between the area of small circles that are tangent to one another and the end circles are tangent to one large circle?  In addition, the extension asks what is the relationship of the circumference of the larger circle to the sum of the circumferences of the enclosed smaller circles?  



Plans to Solve/Investigate the Problem

I am going to use Geometer’s Sketchpad and let it do the “computation” for area and perimeter for me.  I will create several examples using different numbers of tangent circles within a larger circle and see if there is a pattern concerning the area.  Then I will measure the circumferences of the circles and compare to see if there is a pattern or relationship.


Investigation/Exploration of the Problem

I designed four different scenarios:  three tangent circles along a diameter, four circles, five circles and six circles.  The results lead me to believe the following:

There is a pattern to determine the relationship of the area between the combined number of tangent circles along a diameter of a larger circle and the area of the larger circle.  The area of the larger circle will be the same times larger as there are circles.  For example, if you click here, you will notice that if a circle has three tangent circles, the area of the larger circle will be three times the size of the combined areas of the three smaller circles.  Click here to see that the area of a circle with four circles that are tangent along the diameter will be four times the area of the combined four small circles.  Click here to see how it works for 5 smaller circles and click here to see 6 circles. 



Extensions of the Problem

If you were to walk along the entire circumferences of all the small circles, would you walk farther or less far than if you walked around the circumference of the largest circle? How much more or how much less would you walk?

On the same Geometer’s Sketchpad pages, I calculated the circumferences of the different examples.  In all of the examples, the circumferences were equal to the combined circumferences of the smaller circles. 

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