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Tangential Circles

Problem Statement

Two larger circles with equal radii pass through each other's centers. A smaller circle can be created inside the overlapping region so that it is tangent to the other circles. (Tangent means that the circles touch each other but do not cross over each other, nor do they leave any gaps.) Compare the area and circumference of the smaller circle to the area and circumference of the larger circle.


Investigation/Exploration of the Problem

 

For this problem, I did a few quick calculations on a sheet of paper using an arbitrary figure for the radius of the larger circles.  Here is a snapshot of what I came up with using Geometers Sketchpad:

 

Notice that the diameter of the smaller circle is equal to the radius of the larger circles.  Interesting don’t you think?   Well I did!

I digress…. Recall that the formulas for area and circumference of a circle are as follows:

Look in the following math blurp to see what I did with this information.

 

                                             

 

 

 

 

 

Now I truly feel that his would be the case regardless of the length of the diameter of the larger circles.  Let’s investigate using variables:

 

 

 

 

 

 

 

Author & Contact
If you find a flaw in my argument or have questions, feel free to contact me!



 

 

 

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