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Math Help - How much of the area is preserved when you inscribe circles in circles

  1. #1
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    How much of the area is preserved when you inscribe circles in circles




    What is the ratio of the area of the shaded portion in the figure on the right to the area of the shaded portion of the figure to the left? The circles are all tangent to each other even though the picture might not be exact. The inscribed circles in the left figure are congruent. The second figure is a result of inscribing 4 congruent circles in each of the 4 circles on the left.

    I have no idea how to approach this. Thanks for taking the time to read my question!
    Last edited by eeyore; October 5th 2009 at 08:01 PM. Reason: Missed a part of the definition
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  2. #2
    Member pflo's Avatar
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    First, lets just look at the figure on the left with a single circle and four circles inscribed in it. Let A be the area of the larger circle (with radius R) and a be the area of each of the smaller circles (with radius r). You are looking for the following ratio: \frac{A}{4a} which is \frac{\pi R^2}{4(\pi r^2)}=\frac{R^2}{4r^2}. So you essentially need to find the ratio of the radii.

    Put a coordinate system on your circle picture, with the center of the large, original circle at the origin.

    Look at the smaller circle in quadrant I of the coordinate system. It touches both the x-axis and the y-axis. Therefore, its center is at (r,r). Using this, you can calculate the distance from the origin to its center as \sqrt{2}*r and if you go from the center to the point of tangency (adding an r from this point) you will have gone R. This means \sqrt{2}*r+r=R. Solving for r, you can see r=\frac{R}{\sqrt{2}+1}.

    Plug this in the original question and you will find \frac{R}{4r}=\frac{R}{4\frac{R}{\sqrt{2}+1}}=\frac  {\sqrt{2}+1}{4}

    Can you use this information to figure out the answer to your question?
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  3. #3
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    Yes, thank you! That was very helpful.
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