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Math Help - Area of a polygon on the unit circle

  1. #1
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    Area of a polygon on the unit circle

    I'm having a hard time solving this problem:
    Find the area of a regular dodecagon whos vertices are 12 equally spaced points on the unit circle.

    what i did was split the entire polygon into 12 smaller triangles
    A = 1/2absin(theta)
    A = 1/2 (1)(1)sin(pi/4)
    A = pi/4 / 2 = (1/sqrt(2))/2 = 1/2sqrt(2) - for 1 triangle

    for the entire polygon i did 12(1/2sqrt(2)) = 3sqrt(2)
    but then the answer says it's 3
    what am i doing wrong?
    Last edited by dondonlouie; April 5th 2011 at 11:53 PM.
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  2. #2
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    Quote Originally Posted by dondonlouie View Post
    I'm having a hard time solving this problem:
    Find the area of a regular dodecagon whos vertices are 12 equally spaced points on the unit circle. <--- OK

    what i did was split the entire polygon into 12 smaller triangles
    A = 1/2absin(theta)
    A = 1/2 (1)(1)sin(pi/4)
    The central angle of one triangle is calculated by: \frac{2 \pi}{12}=\frac16 \pi.
    ... and \sin\left(\frac16 \pi\right)=\frac12
    A = pi/4 / 2 = (1/sqrt(2))/2 = 1/2sqrt(2) - for 1 triangle

    for the entire triangle i did 12(1/2sqrt(2)) = 3sqrt(2) <--- the value in red is wrong. See above!
    but then the answer says it's 3
    what am i doing wrong?
    ...
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  3. #3
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    Quote Originally Posted by earboth View Post
    ...
    how did you get the central angle? i thought it was pi/4 cuz the two sides each have a radius of 1 D:
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  4. #4
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    Quote Originally Posted by dondonlouie View Post
    I'm having a hard time solving this problem:
    Find the area of a regular dodecagon whos vertices are 12 equally spaced points on the unit circle.

    ...
    what am i doing wrong?
    Quote Originally Posted by dondonlouie View Post
    how did you get the central angle? i thought it was pi/4 cuz the two sides each have a radius of 1 D:
    Make a sketch!

    With 12 points on the unit circle you produce 12 isosceles triangles whose legs include an angle of \frac1{12} \cdot 2 \pi = \frac16 \pi
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  5. #5
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    Quote Originally Posted by earboth View Post
    Make a sketch!

    With 12 points on the unit circle you produce 12 isosceles triangles whose legs include an angle of \frac1{12} \cdot 2 \pi = \frac16 \pi
    ahhh okay i see it now (: thank you so much!
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