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Math Help - Arrangement of circles in a rectangle

  1. #1
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    Question Arrangement of circles in a rectangle

    I have a reactangle of 2.50 x 2.85 m
    I want to put as many cicles of a certain size in this reactangle as possible. How can I calculate (and arrange) those circles?

    Practical case:
    In a rectangular drainage channel i want to put one turbine or a matrix of (smaller but identical) turbines. Radius of turbines can vary from 0.30 m until 1.20 m incremented with 0.05 m.

    I calculated already
    a) 4 x 4 turbines of 0.30 m --> 63% of the area occupied
    b) 3 x 4 turbines of 0.35 m --> 65%
    c) 2 x 2 turbines of 0.60 m --> 63%
    d) 1 turbine of 1.20 m --> 63%

    But I wonder if another arrangement and/or size could use the gaps in between the turbines in the matrices described above (as height is 2.85 m)
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    South Coast of England
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    Hello SoftwareTester
    Quote Originally Posted by SoftwareTester View Post
    I have a reactangle of 2.50 x 2.85 m
    I want to put as many cicles of a certain size in this reactangle as possible. How can I calculate (and arrange) those circles?

    Practical case:
    In a rectangular drainage channel i want to put one turbine or a matrix of (smaller but identical) turbines. Radius of turbines can vary from 0.30 m until 1.20 m incremented with 0.05 m.

    I calculated already
    a) 4 x 4 turbines of 0.30 m --> 63% of the area occupied
    b) 3 x 4 turbines of 0.35 m --> 65%
    c) 2 x 2 turbines of 0.60 m --> 63%
    d) 1 turbine of 1.20 m --> 63%

    But I wonder if another arrangement and/or size could use the gaps in between the turbines in the matrices described above (as height is 2.85 m)
    This is by no means a complete answer, but I just tried arranging 0.30 m radius circles with their centres on a triangular grid, and fitted 18 in - see attached diagram. This increases the coverage to 71%.

    Perhaps you could investigate that further?

    Grandad
    Attached Thumbnails Attached Thumbnails Arrangement of circles in a rectangle-untitled.jpg  
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  3. #3
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    Dec 2007
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    Yes, that's basically the kind of arrangement I had in mind.
    But how do I CALCULATE that?
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