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Math Help - Percentage area of circles overlapping a triangle

  1. #1
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    Percentage area of circles overlapping a triangle

    Hi all,

    I have an interesting problem regarding the area coverage of a triangle, in relation to the diameter of three equal circles centred at each of its three vertices (please see attached image).

    Is it possible to formulate a function that will exact the required circle diameter for a given percentage-coverage of the triangular area? For example, what would be the correct circle diameter for an 89% coverage of the triangular area (i.e. 11% of the triangular area is not obscured).

    As the circles will need to overlap in order to accommodate the complete 0% - 100% range, is it possible to create a unified formula or will the overlapped and non-overlapped percentages need to be addressed separately?

    Any help or insight would be most appreciated.

    Best regards, Gemma.
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  2. #2
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    Quote Originally Posted by Jemma Lovell View Post
    I have an interesting problem regarding the area coverage of a triangle, in relation to the diameter of three equal circles centred at each of its three vertices (please see attached image).

    Is it possible to formulate a function that will exact the required circle diameter for a given percentage-coverage of the triangular area? For example, what would be the correct circle diameter for an 89% coverage of the triangular area (i.e. 11% of the triangular area is not obscured).

    ...
    The 3 sectors of the 3 circles which are in the area of the equilateral triangle form a half circle. Let s denote the side of the triangle then the not obscured area can be calculated by:

    A_{n.o} = A_{triangle} - \frac12 \cdot \pi r^2~,~0 \leq r \leq \frac23 s

    A_{n.o} = \frac14 \cdot s^2 \cdot  \sqrt{3} - \frac12 \cdot \pi r^2~,~0 \leq r \leq \frac23 s

    That means: If you know the length of s and the value of A_{n.o} you can calculate the value of r.
    Attached Thumbnails Attached Thumbnails Percentage area of circles overlapping a triangle-dreieck_kreis.gif  
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  3. #3
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    Hi Earboth,

    Thank you for the reply.

    I think I understand what is going on here, but I still don't see how this works as the coverage approaches 100% (please see attached plot).

    Sorry for being a little dense and thank you for your patience.

    Best regards, Gemma.
    Attached Thumbnails Attached Thumbnails Percentage area of circles overlapping a triangle-plot.gif  
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  4. #4
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    Quote Originally Posted by Jemma Lovell View Post
    Hi Earboth,

    Thank you for the reply.

    I think I understand what is going on here, but I still don't see how this works as the coverage approaches 100% (please see attached plot).

    Sorry for being a little dense and thank you for your patience.

    Best regards, Gemma.
    I'm sorry that I didn't take into account that there will be overlapping areas if the radius is greater than \frac12 s.

    If the radius is 0 \leq r \leq \frac12 s the formula of the previous post works fine. For those cases that \frac12 s < r \leq \frac23 s you have to perform a little bit extra work:

    The complete area which covers the triangle and which is produced by the 3 circles consists of
    6 right triangles (2 of them are painted in orange)
    3 small sectors

    1. Calculate first the length of h: h = \sqrt{r^2 - \frac14 s^2}

    2. Calculate the angle \beta : \tan (\beta) = \frac h{\frac12 s}=\frac{\sqrt{r^2 - \frac14 s^2}}{\frac12 s}~,~0^\circ \leq \beta \leq 30^\circ

    3. Then \alpha = 60^\circ - 2 \beta

    4. Then the area of the triangle which is not covered by any of the circles consists of:

    A_{not\ covered}= A_{triangle} - 6 \cdot A_{right\ triangles} - 3 \cdot A_{small\ sectors} . That means:

    A_{nc} = \frac14 \cdot s^2 \cdot \sqrt{3} - 6 \cdot \frac12 \cdot \frac12 s \cdot \sqrt{r^2 - \frac14 s^2} - 3 \cdot \frac{\alpha}{360^\circ}\cdot \pi r^2~,~\frac12s < r \leq \frac23 s

    5. This is the rough version of the formula. But I believe it is easier to see what's going on and how I got this result if I leave it in this form. It is up to you to simplify this equation a little bit (if possible).
    Attached Thumbnails Attached Thumbnails Percentage area of circles overlapping a triangle-dreieck_ohnekreis.gif  
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