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Math Help - area of a segment of a circle

  1. #1
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    area of a segment of a circle

    Hello!

    How do i get the area of this segment on the top of the circle?
    w=44.5 mm
    h=10.35 mm
    r=34.9 mm

    I have found some formulas for calculating the area, but it seems like I need some angle.. ? What If i dont know the angle...?
    A=0.5*r*(b-w)+w*h
    b is the arc length, b = r*angle*pi/180

    How do I use this angle?
    I know that the answer should be approx 696 mm2 from a computer program, but I cant do it analytically...

    Need help
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  2. #2
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    Erm , from your diagram , i guess W is the length of the chord . If so , we can apply this formula :

     2r\sin\frac{\theta}{2}=44.5 , We know the r and thus we can find \theta

    So now you have the angle .

    Area of a segment = \frac{1}{2}r^2(\theta-\sin\theta)
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  3. #3
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    Quote Originally Posted by mathaddict View Post
    Erm , from your diagram , i guess W is the length of the chord . If so , we can apply this formula :

     2r\sin\frac{\theta}{2}=44.5 , We know the r and thus we can find \theta

    So now you have the angle .

    Area of a segment = \frac{1}{2}r^2(\theta-\sin\theta)

    Thank you!

    The expression 2*r*sin(theta/2)= 44.5. is that a general expression that always can be used or is it just for my case with my chord length ?
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  4. #4
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    Quote Originally Posted by danielel View Post
    Thank you!

    The expression 2*r*sin(theta/2)= 44.5. is that a general expression that always can be used or is it just for my case with my chord length ?
    With the chord w and two radii you have an isoceles triangle. Split it into two right triangles. (see attachment)

    Then you get:

    \sin\left(\frac{\theta}2\right) = \dfrac{\frac12 \cdot w}{r}~\implies~ 2r\cdot \sin\left(\frac{\theta}2\right) = w
    Attached Thumbnails Attached Thumbnails area of a segment of a circle-zentralwinkel.png  
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  5. #5
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    Thumbs up

    Thank you!
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