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Length of a line between origin and edge of a circle

  1. Nov 8, 2015 #1
    For a circle with radius R centered at R along the X-axis so that the edge of the circle touches the origin, what is the length of a line drawn between the origin and an edge of the circle in terms of the angle between that line and the X-axis? This isn't a homework problem, just something I'm trying to figure out.

    IE in this picture (sorry about the MS Paint, I'm on a fairly old computer right now), what is the length of the line L in terms of the angle a and radius R? Any help is much appreciated, thank you.

    fxIEn7b.jpg
     
  2. jcsd
  3. Nov 8, 2015 #2

    fresh_42

    Staff: Mentor

    Are you firm in computing sides of triangles?
     
  4. Nov 8, 2015 #3

    Yes, I'm pretty comfortable with that. What are you suggesting?
     
  5. Nov 8, 2015 #4

    fresh_42

    Staff: Mentor

    There are 2 possible triangles you can use to compute L depending on a. One involves Thales' Theorem. But it also can be done without. Just look for triangles and right angles you can find in your graphic.
     
  6. Nov 8, 2015 #5

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    This article shows the trigonometry of working out L based on a circle with unit radius:

    http://geowords.com/e_/06_wind&compass/pi/protractor.htm

    Scroll down to 2) at the link. The diagram there:

    chord.gif
    Matches what you drew in the OP.

    Taking the diameter of an arbitrary circle as D, which is the length of EB in the picture above, then cos α = AE / EB, or using your diagram, cos α = L / D.

    Therefore, L = D cos α

    or L = 2R cos α
     
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