Friday, April 04, 2008

Hitting the Target Puzzle

Target DiagramHere's a quick puzzle. In the attached image, a circle is inscribed in a square which is inscribed in another circle.

Of the outside yellow ring, or the inside magenta circle, which has the bigger area, and why?

5 comments:

  1. Imagine the square root of 2.

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  2. Well, the inner circle has an area of pi*radius**2.

    The radius of the larger circle is a function of the hypotenuse of the square. The hypotenuse(RADIUS**2) of the square is (2*radius)**2 + (2*radius)**2

    So, the area of the outer circle is pi*8*radius**2. Subtract the inner circle area and you get pi*7*radius**2 for the outer area, which is greater than the area of the inner circle.

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  3. Don, double-check your math. The yellow ring does NOT have 7 times the area of the magenta circle.

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  4. Yes, I left an extra '2' in the equation. Shame.

    The areas are equal.

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  5. Well done, all!

    Solution diagram

    The key is the square. Looking at the two circles, the ratio of their radii will be sqrt(2) : 1 because they are the hypotenuse and leg of a 45-45-90 triangle).

    The circles will therefore have areas in the ratio of 2 : 1. When you subtract the inner circle to leave just the ring, you have two areas that are equal.

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