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Math Help - Can I calculate the radius of a circle from arc length and arc height alone?

  1. #1
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    Question Can I calculate the radius of a circle from arc length and arc height alone?

    Hi there

    I have a problem which I thought would be quite easy when I was out on site.
    I have a curved bridge which I am assuming is part of a circle.

    I measured the arc length AL and by pulling a string taught and dropping a ruler down measured the distance from the centre of the arc to the chord which I call h.

    I thought that these two measurements must define an individual circle, and did not make an attempt to measure the chord length.

    I am now trying to define the radius and angle etc. of the arc but cannot do this algebraically. I could probably attack this iteratively to find the particular solution using excel but I'd like to know if there is a direct equation to solve this.

    I have been through a few sides of A4 trying to solve this and suspect I do not have enough information, so here is some of my equations I have written but cannot see how to combine them to get r the radius or theta alone.

    r = radius
    theta = half the included angle
    AL = arc length
    CL = chord length
    h = Arc Height ( max distance from chord to Arc)


    r*cos (theta) = r-h
    CL = 2*r*sin( theta )
    AL = r * theta
    r^2 = (CL/2)^2 + (r-h)^2

    What approach should I be taking?

    (I have found an online calculator that has given me a result but I am frustrated with my own attempts to solve it directly)

    Any help would be most appreciated.

    Thanks

    Matt
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  2. #2
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    Re: Can I calculate the radius of a circle from arc length and arc height alone?

    This is a classic problem that goes back to the Babylonians and is called the "bow and arrow" problem. (The arc of the circle and chord are the bow and bowstring. The radius through the center is the arrow.)

    Call the length of the chord "l" and the height, from chord to arc at the center of the arc, "h". Call the (unknown) radius of the circle "r". The distance from the center of the circle to either end of the arc is r, the radius. The distance from the center of the circle to the center of the chord is r- h. The distance from the center of the chord to one endpoint is l/2. Those lines form a right triangle so, by the Pythagorean theorm, (r-h)^2+ l^2/4= r^2 which, multiplying that first square, is r^2- 2rh+ h^2+ l^2/4= r^2 and then, canceling the " r^2" terms, reduces to -2rh+ h^2+ l^2/4= 0. We can write that as 2rh= h^2+ l^2/4 and so r= \frac{h^2+ \frac{l^2}{4}}{2h}= \frac{4h^2+ l^2}{8h}.
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  3. #3
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    Re: Can I calculate the radius of a circle from arc length and arc height alone?

    Hi there

    Thanks for the quick reply.

    I may have confused Arc length and the chord length in my initial statement. I have the measurement around the outside of the arc and the height from the chord to the arc not the chord length itself.

    Unfortunately I did not measure the chord length on site assuming I had enough information already to cal the rest. I only have the arc length, and the height. I have been trying to calculate the chord length from the arc length and height alone. Surely there is only one chord length for a given height and arc length ? When I can calculate the chord length I can then use the equation you've given me (nice explanation by the way) to get the radius and the rest.

    Am I missing something startlingly obvious perhaps to do with triangles within circles?

    Thanks again

    Matt
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    Re: Can I calculate the radius of a circle from arc length and arc height alone?

    Quote Originally Posted by hairymarmite View Post
    Unfortunately I did not measure the chord length on site assuming I had enough information already to cal the rest. I only have the arc length, and the height. I have been trying to calculate the chord length from the arc length and height alone. Surely there is only one chord length for a given height and arc length ? When I can calculate the chord length I can then use the equation you've given me (nice explanation by the way) to get the radius and the rest.

    Am I missing something startlingly obvious perhaps to do with triangles within circles?
    Have a look at this page.
    You can see how a change in radius changes cord length.
    There are many formulas there that may help you.
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  5. #5
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    Re: Can I calculate the radius of a circle from arc length and arc height alone?

    I think I have convinced myself, that I cannot define a single circle from the two figures alone.

    The chord length measurement would fix the ends of thew arc a certain distance apart and the radius and the angle can then be calculated.
    I can imagine that a large radius arc would have a small angle and give both the same height and arc length as a smaller radius arc with a larger angle would.

    I've pictured it in my head but haven't done the math and it makes sense, though I now have a head ache.


    Plugging equations into each other gave me the following (unless I've made a mistake somewhere)

    cos theta = 1 - h/R
    since arc length AL = 2 * R * theta
    I wrote
    cos ( AL/2R ) = 1 - h/R


    thanks for the pointers
    let me know if I am wrong about this.

    Matt
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  6. #6
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    Re: Can I calculate the radius of a circle from arc length and arc height alone?

    Hi Matt,

    The arc of a bridge span is a catenary curve
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  7. #7
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    Re: Can I calculate the radius of a circle from arc length and arc height alone?

    Since this is a couple of years old, did you find any solutions to this? Working on end contraction from catenary deflection, usually do process stuff. So far this is what it looks like to me.

    There is only one radius for any given ratio of arc length to truss height provided the arc is a circle. It is just non-factorable.
    S = arc length
    theta in radians
    h is the sagitta height
    h/S = (1-cos(theta/2))/theta had to iterate it out....used a while wend loop.
    then the chord length is (4h*(2S/theta-h)^0.5
    radius is S/theta
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  8. #8
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    Re: Can I calculate the radius of a circle from arc length and arc height alone?

    Hi hazenvr
    Thanks for your reply.


    I am no great mathematician though can fight with equations when required.

    I eventually figured that there must be a single solution, but could not directly calculate it.

    I think I did some sort of butured newton-raphson to give me the answer to an awful lot of decimal points, the result actually matched an online calculater I found exactly so assumed they must have used a similair method . When I plugged that in as the radius it all came out right.
    As it turned out when I took my numbers back on to site my they were not accurate enough to the curve in question, I would need them to be within 1/8 of an inch for me to build the structure in the workshop and reassemble on site. So either my assumption that the steel work was circular was incorrect, or the bridge had sagged over 60 years, or corrosion had deformed the top surface just enough.

    So I borrowed my mates surveying laser which got me within 5mm every meter which I had to live with and built it in the workshop cutting the topcurves to suit so that every single stantion could be the same length.

    Can I calculate the radius of a circle from arc length and arc height alone?-curved200x150.jpg

    It worked out ok. And has been submerged a few times since.

    It's the sort of thing that would irritate me not knowing i there was some way to directly get the answer, but I had to get the job done and move on to the next problem...

    Cheers for your insight though

    Matt
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