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Math Help - Finding a Shaded Region Using the Arcs of Circles

  1. #1
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    Finding a Shaded Region Using the Arcs of Circles

    I have a geometry problem, which I believe I have worked out but seems way too easy.

    The question describes a scenario where arcs of circles are contained within a square of length A. Therefore, its area is A^2. The requirement is to work out the area of a shaded region created by these arcs in terms of A.

    A perfect circle is formed within the square. Because the length of the square is A, the radius of the circle within it can be defined as 1/2 of A.

    Therefore the area of the circle is simply Pi x (1/2 A)^2 = Pi(1/4 x A^2).

    The area of the square that does not include the circle is the area of the square minus the area of the circle:

    A^2 - Pi(1/4 x A^2)

    But from looking at the diagram and considering its symmetry, I can tell that each corner forms an arc with a radius of 1/2 of A, just like the central circle.

    Consequently, these arcs forms 4 quarter-circles, which all combine to create an area equal to the main circle.

    Because these quarter-circles include all the shaded regions, I can use their combined area and subtract from it the area of the square not contained in the circle:

    Pi(1/4 x A^2) - (A^2 - Pi(1/4 x A^2)) = 2Pi(1/4 x A^2) - A^2

    Therefore, 2Pi(1/4 x A^2) - A^2 expresses the area of the shaded region/s in terms of A.

    I have included a link to a rough MS Paint diagram I drew to illustrate the problem. The diagram is meant to be symmetrical - please disregard the slight asymmetry of the arcs.

    http://www.facebook.com/photo.php?pid=30017852&l=4d468&id=1206742428

    I am wondering - have I gone about this problem correctly? Once again, it seems far too simple. Is there another way to do it that I have not used?

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by happy_go_lucky View Post
    I have a geometry problem, which I believe I have worked out but seems way too easy.

    The question describes a scenario where arcs of circles are contained within a square of length A. Therefore, its area is A^2. The requirement is to work out the area of a shaded region created by these arcs in terms of A.

    A perfect circle is formed within the square. Because the length of the square is A, the radius of the circle within it can be defined as 1/2 of A.

    Therefore the area of the circle is simply Pi x (1/2 A)^2 = Pi(1/4 x A^2).

    The area of the square that does not include the circle is the area of the square minus the area of the circle:

    A^2 - Pi(1/4 x A^2)

    But from looking at the diagram and considering its symmetry, I can tell that each corner forms an arc with a radius of 1/2 of A, just like the central circle.

    Consequently, these arcs forms 4 quarter-circles, which all combine to create an area equal to the main circle.

    Because these quarter-circles include all the shaded regions, I can use their combined area and subtract from it the area of the square not contained in the circle:

    Pi(1/4 x A^2) - (A^2 - Pi(1/4 x A^2)) = 2Pi(1/4 x A^2) - A^2

    Therefore, 2Pi(1/4 x A^2) - A^2 expresses the area of the shaded region/s in terms of A.

    I have included a link to a rough MS Paint diagram I drew to illustrate the problem. The diagram is meant to be symmetrical - please disregard the slight asymmetry of the arcs.

    http://www.facebook.com/photo.php?pid=30017852&l=4d468&id=1206742428

    I am wondering - have I gone about this problem correctly? Once again, it seems far too simple. Is there another way to do it that I have not used?

    Thanks in advance.
    Your solution is correct. To summarize: The white area inside the big circle is the same as the white area outside the circle but inside the square, which is easily computed as B = A^2 - \pi\frac{A^2}{4}. Then the shaded area is equal to the area of the circle - the white area inside the circle, or \pi\frac{A^2}{4} - B, which is the same as your expression.
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