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Finding the Area of a polygon

  1. May 21, 2007 #1

    disregardthat

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    Hi, I am to find a formula for the area of a regular polygon with a side "a".

    I just keep getting the wrong answer: this is how i did it:

    if we draw a circle in a coordinate system, with radius "r". The diameter lyes on the x-axis. I draw an angle from the center. This angle is then 360/n where n is the amount of sides the polygon can have.

    The two other angles in the triangle we get with two sides "r" and one side "a" is 180/n.

    Ok, to find the side r expressed with a:

    [tex]r^2 = r^2 + a^2 2ra \cos{\frac{180}{n}}[/tex]

    [tex]a = 2r \cos{\frac{180}{n}}[/tex]

    [tex]r = \frac{a}{2 \cos{\frac{180}{n}}}[/tex]

    The area of this triangle is:

    [tex]A = \frac{1}{2} \sin{\frac{180}{n}} ar = \frac{1}{2} \sin{\frac{180}{n}} \frac{a}{2 \cos{\frac{180}{n}}} a = \frac{1}{4} \tan{\frac{180}{n}} a^2[/tex]

    The area of the whole polygon will then be the area of the triangles in the circle. I multiply with the number I divided 360 with, "n".

    So: [tex]A_n =\frac{n}{4} a^2 \tan{\frac{180}{n}} [/tex]

    But this is wrong! Why is it wrong?

    The correct answer is:
    [tex]A_n =\frac{na^2}{4 \tan{\frac{180}{n}}}[/tex]
     
    Last edited: May 21, 2007
  2. jcsd
  3. May 21, 2007 #2
    [tex]r^2 = r^2 + a^2 - 2ra \cos{\frac{180}{n}}[/tex]

    This is wrong. The angle is 90 - 180/n, hence giving [tex]r^2 = r^2 + a^2 - 2ra \sin{\frac{180}{n}}[/tex]
     
    Last edited: May 21, 2007
  4. May 22, 2007 #3

    uart

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    I've got no idea where that line came from but it looks wrong (edit: ok I now see it was supposed to be the cosine rule). You should have just used :

    [tex] a/2 = r \sin(180/n)[/tex]

    Which gives : [tex]r = \frac{a}{2 \sin(180/n)}[/tex]

    Now just substitute that into :

    [tex]A = n ( \frac{1}{2} r^2 \sin(360/n) )[/tex]

    PS. Remember to use the trig identity : [tex]\sin(2x) = 2 \sin(x) \cos(x)[/tex] if you want to get your answer in exactly the same form as the one given.
     
    Last edited: May 22, 2007
  5. May 22, 2007 #4

    disregardthat

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    Yes, it was the cosine rule I meant.

    Hmm, that was wierd. We are not supposed to use trigonometric identities. Or at least the book doesn't mention any of it.
     
  6. May 22, 2007 #5
    Well, uart expression is equivalent to [tex]A = n r^2 \sin(180/n)\cos(180/n) [/tex]
     
    Last edited: May 22, 2007
  7. May 22, 2007 #6

    uart

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    You can get a perfectly good (correct) answer without even using that last trig idenity, it just wont be in the exact same form as the one given. It will be 100% equivalent but just not an identical form.
     
    Last edited: May 22, 2007
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