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Math Help - Finding the internal angles of a polygon

  1. #1
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    Finding the internal angles of a polygon

    Hi there,

    This might seem a simple question, but I suck at maths. I need to cut some ply to fill a gap behind my boat kitchen and I have measured the lengths of the sides. I'd like to have the angles to help me draw a decent template to cut from, but I'm not sure how.

    Finding the internal angles of a polygon-img_0001.jpg

    A = 1055mm
    B = 399mm
    C = 860mm
    D = 73mm

    And the angle between D and A is a right angle.

    Any help welcome.

    Thanks.
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  2. #2
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    You'll need to draw a line from the vertex between B and C, to the vertex between A and D.

    You will then need to use a combination of the sine and cosine rules to find the missing sides and angles...
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  3. #3
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    It would help if you are working with triangles.

    First draw a line from angle AB to angle CD. This will divide the shape into 2 triangles.
    Solve this unknown length (call this E) by using the Pythagorus Theorum (Because it has a right angle):

    A^2 + D^2 = E^2 (I substituted your letters into it)

    1055^2 + 73^2 = E^2

     E \approx 1057.52\mbox{mm}

    Now we find the angle of BC by using the cosine rule:


    \mbox{angle BC} = cos^{-1}\left(\dfrac{B^2+C^2-E^2}{2BC}\right)

    \mbox{angle BC} = cos^{-1}\left(\dfrac{399^2+860^2-1057.52^2}{2 * 399 * 860}\right)

    \mbox{angle BC} \approx 108.657^{\circ}


    To solve the angle AB, we do the same thing, but must divide it into 2 parts.

    \mbox{angle AE} = cos^{-1}\left(\dfrac{A^2+E^2-D^2}{2AE}\right)

    \mbox{angle AE} = cos^{-1}\left(\dfrac{1055^2+1057.52^2-73^2}{2 * 1055 * 1057.52}\right)

    \mbox{angle AE} \approx 3.958^{\circ}

    \mbox{angle EB} = cos^{-1}\left(\dfrac{E^2+B^2-C^2}{2EB}\right)

    \mbox{angle EB} = cos^{-1}\left(\dfrac{1057.52^2+399^2-860^2}{2 * 1057.52 * 399}\right)

    \mbox{angle EB} \approx 50.3976^{\circ}

    \mbox{angle AB} = \mbox{angle AE} + \mbox{angle EB}

    \mbox{angle AB} = 3.958^{\circ} + 50.3976^{\circ}

    \mbox{angle AB} \approx 54.3556^{\circ}


    To find angle CD, we must subtract the other 3 angles by 360.

    \mbox{angle CD} = 360^{\circ} - 54.3556^{\circ} - 108.657^{\circ} - 90^{\circ}

    \mbox{angle CD} \approx 106.9874^{\circ}
    Last edited by Educated; August 15th 2010 at 01:47 AM.
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  4. #4
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    That's fantastic. Thank you very much.
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