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Thread: Calculating the radius of a circular segment

  1. #1
    mklotz mklotz's Avatar
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    Calculating the radius of a circular segment

    The law of cosines, apparently not well known, was used in a previous post...

    Stepping off a polygon

    to good effect to solve a not uncommon shop problem.

    Here we use a somewhat more obscure theorem to unravel another shop problem.

    Suppose you have a circular segment like the figure APBXA in the illustration below. It could be a gear fragment, a key segment you want to reproduce, or even a flower bed you want to extend. How do you go about determining the radius of the circle from which the segment derives?




    The tool we'll use is the intersecting chords theorem. It says that, if two chords intersect inside a circle, their individual parts are mathematically related to each other. A full description, including a proof of the theorem, is here...

    https://en.wikipedia.org/wiki/Inters...chords_theorem


    From the intersecting chords theorem, we have:

    XP x XQ = XA x XB

    Define:

    c = AB ; c = chord length of segment
    h = XP ; h = "height" of segment

    which are things we can measure from the segment, then, substituting into the intersecting chords theorem, we have:

    (c/2)^2 = h x XQ

    which can be solved for XQ

    XQ = c^2/(4h)

    The diameter, PQ = PX + XQ and

    PX + XQ = h + c^2/(4h) = (4h^2 + c^2)/(4h)

    and the radius of the circle, CQ, is half this diameter

    r = CQ = (4h^2 + c^2)/(8h)
    ---
    Regards, Marv


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  2. The Following 7 Users Say Thank You to mklotz For This Useful Post:

    Frank S (05-28-2018), Jon (05-28-2018), natie123 (06-18-2018), PJs (05-29-2018), rossbotics (05-30-2018), Seedtick (05-28-2018), Tuomas (05-29-2018)

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    Imabass's Avatar
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    Excellent. This is common problem for buying the correct fenders for motorcycle.

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    Thanks mklotz! We've added your Circular Segment Radius Calculation Method to our Measuring and Marking category,
    as well as to your builder page: mklotz's Homemade Tools. Your receipt:




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    Excellent, Marv . . . keep them coming! Your timing is perfect . . . I need to extend the driveway. :-)

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    mklotz mklotz's Avatar
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    The radius calculation here can also be used to find a circle that will pass through three (non-colinear) points.

    If you have the three points identified as A, B, and P in the diagram, the calculation will, as described, find the radius of the circle that passes through those points.

    Finding the center of that circle is trivial. Using the calculated radius, draw a circle around two of the points. One of the intersection points of those circles is the center of the circle passing through all three points. It should be obvious which it is but, if unsure, draw a circle around the remaining point; where all three circles intersect is the center.
    Last edited by mklotz; 05-31-2018 at 11:28 AM.
    ---
    Regards, Marv


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    Often needed for fire box to cook chamber on a smoker...

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    mklotz mklotz's Avatar
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    Imabass, Saltfever and GreenSmoker

    Thanks for your comments indicating that this mathematical exercise has some real utility for you.

    So often, folks are completely put off by anything that even hints of math, that its utility is overlooked as they flee. Showing them that there are applications out there where it can help will encourage them to look a bit closer at the next offering.


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    Regards, Marv


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