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Math Help - Radius from Arc

  1. #1
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    Radius from Arc

    Please Help

    If the arc length s and the distance h from the centre point of the associated chord to the arc is known, how do you calculate the radius?

    Thank You
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  2. #2
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    Quote Originally Posted by askmemath
    Please Help

    If the arc length s and the distance h from the centre point of the associated chord to the arc is known, how do you calculate the radius?

    Thank You
    I don't think you will find a closed form solution for this problem - though
    I am prepared to be proven wrong.

    RonL
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  3. #3
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    Here is a picture to help you out.
    By the theorem of chord in a circle we have that,
    n^2=r(2r-h)
    But, n=r\sin(s/2r)
    Thus, we have,
    r^2\sin(s/2r)=r(2r-h)[/tex]
    Thus,
    r\sin(s/2r)=2r-h
    ----------------------
    If you know calculus
    Thus, you need to find the zero's of the function,
    f(x)=x\sin(s/2x)-2x+h
    Use Newton's method
    Attached Thumbnails Attached Thumbnails Radius from Arc-picture7.gif  
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  4. #4
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    Quote Originally Posted by ThePerfectHacker
    Here is a picture to help you out.
    By the theorem of chord in a circle we have that,
    n^2=r(2r-h)
    Just as well that I checked my note on this problem

    The intersection chord theorem would give in this case:

    <br />
n^2=h(2r-h)<br />

    surly?

    RonL
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  5. #5
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    Quote Originally Posted by CaptainBlack
    Just as well that I checked my note on this problem

    The intersection chord theorem would give in this case:

    <br />
n^2=h(2r-h)<br />

    surly?

    RonL
    One thing that confuses me is how an immortal (me) makes such a mistake
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  6. #6
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    Quote Originally Posted by ThePerfectHacker
    Here is a picture to help you out.
    By the theorem of chord in a circle we have that,
    n^2=r(2r-h)
    But, n=r\sin(s/2r)
    Thus, we have,
    r^2\sin(s/2r)=r(2r-h)
    ...
    Hello,

    I'm a little bit confused: When you plug in the value of n, shouldn't be there a squared sine value too?:

    r^2\left(\sin(s/2r)\right)^2=r(2r-h)

    Greetings

    EB
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by earboth
    Hello,

    I'm a little bit confused: When you plug in the value of n, shouldn't be there a squared sine value too?:

    r^2\left(\sin(s/2r)\right)^2=r(2r-h)

    Greetings

    EB
    There should be, now how did I miss that (its in my notes)

    RonL
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  8. #8
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    Quote Originally Posted by ThePerfectHacker
    One thing that confuses me is how an immortal (me) makes such a mistake

    Oh, PLEASE!

    don't gamble with your rule nr. 5

    EB
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