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Thread: Finding a Chord Length of a Circle if I have Arc Length and Height?

  1. #1
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    Finding a Chord Length of a Circle if I have Arc Length and Height?

    I am in the process of making a tent and need to get two curved surfaces to meet.
    I need to find the length of a Chord of a circle given that I have the Arc length and Arc Height (that's all), no radius or anything else.

    I suspect that I will need a radius to find this. Or am I missing a point here?

    The below spreadsheet I found on the web proports to be able to do this but the answer is not correct. The edges of my material does not match up when I calculate the Chord length given that I've entered the Arc Length and the Arc Height.

    Does anyone know how to do this?
    Thanks

    http://mathforum.org/dr.math/gifs/ChordMath.xls
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  2. #2
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    Re: Finding a Chord Length of a Circle if I have Arc Length and Height?

    Call the known chord height 'H' and the known arc length 'A'. If we define R as the radius of the arc and theta as the half angle for the arc length, we have two equations in two unknowns (R and theta):

    $\displaystyle R\cos \theta + H = R$

    $\displaystyle R \theta = A $

    Combine and rearrange to get:

    $\displaystyle A \cos \theta + 2 H \theta = A $

    I don't believe there is a closed-form solution for this, so you will have to solve for theta using a numerical technique. The radius can then be found using $\displaystyle R = \frac A {\theta} $, and then the chord length will then be $\displaystyle 2 R \tan \theta$.
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  3. #3
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    Re: Finding a Chord Length of a Circle if I have Arc Length and Height?

    the spreadsheet you referenced seems to give the correct answer.

    what values did you try?
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  4. #4
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    Re: Finding a Chord Length of a Circle if I have Arc Length and Height?

    Your question was asked, answered and extensively discussed here.

    Finding Chord length given arc length and arc height? - My Math Forum

    In particular the veracity of the spreadsheet was confirmed.
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