Tangent Along the Diameter

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?

This problem was found on the Intermath site at:

http://www.intermath-uga.gatech.edu/topics/geometry/circles/r03.htm

Click here to view the GSP file to change the size of the circles (move point B).

The most obvious operation to apply here to find the relationship between the sum of the areas of the smaller tangent circles to the area of the larger circle, is to divide the area of the larger by the sum of the smaller areas. In changing the size of the circles, it was very consistently approximately 3. It is probably not exactly 3 every time due to rounding of the value of pi. This should be obvious from the given picture since each smaller circle has a diameter equal to one-third of the diameter of the larger circle. In the following "proof," A sub L represents the area of the large circle and A sub S represents the area of the small circles.

In making the connection above, it now seems very obvious that the sum of the four small tangent circles along the diameter of the large circle should be 1/4 of the area of the large circle. Again, we have small circles that have a diameter equal to 1/4 of the diameter of the large circle.

Click here to view the GSP file to change the size of the circles (move point I).

Therefore, our general case has a tangent circles along the diameter of a larger tangential circle. The sum of the areas of the smaller circles equals 1/a times the area of the larger circle.

From the Intermath site, there is also an extension activity related to this problem involving the circumferences of the circles.

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?

Is the sum of the circumferences of the smaller tangent circles always equal to the circumference of the large tangent circle?

I have inserted two spreadsheets so that we can examine this by looking at several values at one time.

It appears as if the sum of the circumferences of the small tangential circles is equal to the circumference of the larger tangential circle. Therefore, you are walking no further nor no less.

Now, our fun must come to an end. Not to worry, there are more assignments that will allow us to frolic as we have done today!

Just wanted to leave you with another image of real life tangent circles.

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