1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding the Area of a polygon

  1. May 21, 2007 #1

    disregardthat

    User Avatar
    Science Advisor

    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

    User Avatar
    Science Advisor

    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

    User Avatar
    Science Advisor

    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

    User Avatar
    Science Advisor


    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finding the Area of a polygon
  1. Find the area (Replies: 2)

Loading...