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Math Help - Construction Rectangle in Circle

  1. #1
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    Construction Rectangle in Circle

    Given a circle and a square next to the circle . How can I construct a rectangle with the same surface as the square in the circle? Only using a compass and ruler.

    Thanks
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    Measure the length of one side of the square.

    The length of the rectangle will be that length, x, multiplied by a certain constant k you don't know.

    So, l = kx
    The width will be w = x/k

    So that the area of the square is x^2 and the area of the rectangle is (kx)(x/k) = x^2

    In your circle of radius r that you measure, you will find that:

    l^2 + w^2 = (2r)^2

    Substitute l and w.

    (kx)^2 + \left(\dfrac{x}{k}\right)^2 = 4r^2

    Plug in the values of x and r to find k.

    From there, you can get l and w, the dimensions of the rectangle to be drawn in your circle.
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  3. #3
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    Thanks Unknown008. But I ment a ruler without numbers. So this should be a question in the topic "Euclidean Geometry". Can you help?
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  4. #4
    MHF Contributor Unknown008's Avatar
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    Oh, then no, sorry, I don't know about Euclidean Geometry... I'd be most happy to help otherwise.
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  5. #5
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    Quote Originally Posted by bram kierkels View Post
    Given a circle and a square next to the circle . How can I construct a rectangle with the same surface as the square in the circle? Only using a compass and ruler.

    Thanks
    I think the solution is that "squaring the circle" is impossible...
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  6. #6
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    the square is the maximum area quadrilateral which can be inscribed in a circle so the problem is impossible


    bjh
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  7. #7
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    The original post doesn't give any values for the diameter of the circle
    and the side of the square.
    The circle certainly needs to be of a minimum size in relation to the square.

    Also, may we assume "surface" refers to surface area ?
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  8. #8
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    construction of rectangle in circle

    Followup to prior post

    Given a square side lenght a and a circle diameter b.
    the square when inscribed in a circle produces the largest area of the 4sided polygons.
    A square with side a can be inscribed in a circle with diameter a*rad2. No rectangle with area a^2 can fit in this circle.An infinite number of smaller area rectangles will fit with areas ranging from close to 0 to a max close to a^2.These rectangles have diagonal lengths a*rad2


    bjh
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