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Math Help - geometry/algebra. Areas with only perimeter given

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    geometry/algebra. Areas with 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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    Hello,

    Quote Originally Posted by ttlpkg32 View Post
    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
    For someone who doesn't know how to post, I find it wonderful that your title is so clear !

    Ok, let's go for it.

    Let x be the side of the square (see picture below).
    Let's calculate the perimeter in respect with x.

    We know that each semicircle is 15cm.

    But ! You know that they are semicircles of a circle which diameter is x.
    As the perimeter of a semicircle is half the perimeter of the circle :

    15=\frac 12(2 \pi \frac x2)

    15=\frac{\pi x}{2}

    Hence \boxed{x=\frac{30}{\pi}}

    ~~~~~~~~~~~~

    Now, the area !

    The area is the sum of the areas of :
    - the square
    - the 4 semicircles.

    You should know the area of the square

    So let's see what the area of 1 semicircle is :

    The area of a semicircle is half the area of the circle.
    Hence A_s=\frac 12 (\pi r^2), where r is actually \frac x2

    ---> \boxed{A_s=\frac 12 (\pi \left(\frac x2 \right)^2)=\frac{\pi x^2}{8}}



    So what is the area of the whole thing ?
    Attached Thumbnails Attached Thumbnails geometry/algebra. Areas with only perimeter given-square.jpg  
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