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Math Help - Calculation for how many rectangles fit in a circle

  1. #1
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    Calculation for how many rectangles fit in a circle

    Hi,
    I am trying to optimise how many rectangles I can fit into a circle. The diameter of the circle is always 1100mm but the size of the rectangles varies, however, I would only ever want to pack one size rectangle into a circle at any one time.

    I was wondering if there is an equation to work out the number of rectangles that can fit into the circle, when the dimensions of the rectangle (ie width and depth) and circle (diameter) are both known?

    I have looked on the internet but I am yet to find a comprehensive answer.

    I look forward to your thoughts.

    Ross
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  2. #2
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    Quote Originally Posted by RossT View Post
    Hi,
    I am trying to optimise how many rectangles I can fit into a circle. The diameter of the circle is always 1100mm but the size of the rectangles varies, however, I would only ever want to pack one size rectangle into a circle at any one time.

    I was wondering if there is an equation to work out the number of rectangles that can fit into the circle, when the dimensions of the rectangle (ie width and depth) and circle (diameter) are both known?

    I have looked on the internet but I am yet to find a comprehensive answer.

    I look forward to your thoughts.

    Ross
    Gauss had worked out a formula for the number of lattice points in a circle. What you are asking is about the same. The size of your squares/rectangles would determine the lattice points or corners.

    How critical is it to get the maximum number of rectangles inside the circle?

    This is starting to look like a packing problem. There are too many variables (unknowns) for a unique solution. Each size rectangle will have an area that CANNOT be covered -- you want to minimize that "loss".

    My first attempt would be to ONLY use the first quadrant (then multipy by 4).
    Start at the origin and place rectangles along the x-axis until it touches the edge of the circle.
    Stack the next set of rectangle atop the first set.
    etc.
    At some point you will have an area left over that MAY allow the placement of more rectangles.

    For specific rectangle dimensions, there is a maximum you can fit inside the circle.
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