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Math Help - Semicircles/Square but only perimeter given

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    Semicircles/Square but only perimeter given

    I am having trouble figuring out where to begin.
    The problem:

    A figure has the shape of a square with semicircles on each side. The figure has a perimeter of 60ft. Find the area of the figure.

    I am thinking that the arc of eac semicircle which would be that perimeter of each is 15 ft. But that is as far as I can get. I am frustrated. haha

    Bill
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    Quote Originally Posted by ttlpkg32 View Post
    ...

    A figure has the shape of a square with semicircles on each side. The figure has a perimeter of 60ft. Find the area of the figure.

    ...
    1. Draw a sketch

    2. The perimeter of the figure is:

    p = 2x+2 \pi \cdot \frac x2 = x(2+\pi)

    3. The perimeter is 60' :
    60 = x(2+\pi) ~\implies~ x=\frac{60}{2+\pi}

    4. The area of the figure is a square and a circle:

    A = x^2 + \pi \cdot \left(\frac x2\right)^2 = x^2\left(1+\frac{\pi}4\right)

    5. You should get: A \approx 243.132\  squft
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    From the original post, sounds like it's a square with 4 semicircles attached - one on each side - not just two. Since the perimeter is 60, the OP's assumption that each semicircle has a perimeter of 15.

    Now, take two of those semicircles, put them together, and you have a full circle with a perimeter of 30. Remember that C = \pi d, so the diameter of each circle is \frac{30}{\pi} - thus, the radius is \frac{15}{\pi} - we'll need both the diameter and the radius for this.

    Now, we have two full circles in our shape (4 semicircles), so they have a TOTAL area of 2\pi r^2. Putting in our radius gives:

    2 \pi (\frac{15}{\pi})^2 = \frac{450}{\pi}.

    So that's the area of our four semicircles. Now, note that the square has a side length that's equal to the diameter of the circle, which we said was \frac{30}{/pi}. To find the area of a square, just square the side length. So, the area of our square is:

    A = \frac{900}{\pi^2}.

    To find the total area of both shapes, add them together:

    TA = \frac{450}{\pi} + \frac{900}{\pi^2} = \frac{450\pi + 900}{\pi^2} \approx 234.43
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