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Arc Length of a Circle

  1. Mar 19, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the Arc Length from (0,3) clockwise to (2,sqrt(5)) along the circle defined by x2 + y2 = 9


    2. Relevant equations
    Arc Length formula for integrals

    3. The attempt at a solution
    I have the correct answer at 3arcsin(2/3), but I tried to do this without calculus the first time using the formula s = (r2θ)/2 but I seem to have lost what I once knew from geometry.
    I used the vectors u = <0,3> and v = <2,sqrt(5)> by the points I was given and the origin (0,0) I used the formula cos(θ) = ( u . v )/ ( ||u|| ||v||) the . here denotes the dot product. Solving for theta I have θ=cos-1(u.v)/(||u|| ||v||) and I somehow ended up with an answer roughly 90 times as large. I know it's something frustratingly basic I've mis-remembered or screwed up here. But i'm not sure what it is. Thanks in advance.
     
  2. jcsd
  3. Mar 19, 2013 #2

    Simon Bridge

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    You want to define a small length element dl on the circle at position (x,y) ... changing to polar coordinates would help here.
    Without calculus, the arc length is given by ##s=r\theta##
     
  4. Mar 19, 2013 #3
    if I let x = rcos(θ) and y = rsin(θ) I have r2(cos2θ+ sin2θ) = 9 and r = 3. if I take dr/dθ and square it I get 1. for the bounds, I believe I have 48.2° ≤ θ ≤ 90°.
     
  5. Mar 19, 2013 #4

    SammyS

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    If you want the arc length geometrically (perhaps trigonometrically would be better terminology) then just use
    Arc length = R(θ21) ,

    where θ2 = π/2

    and θ1 = arccos(2/3)​
     
  6. Mar 19, 2013 #5
    I'm getting 2 and some change.
    thanks.
     
    Last edited: Mar 20, 2013
  7. Mar 19, 2013 #6

    SteamKing

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    And make sure you use radian measure. Degrees won't work.
     
  8. Mar 19, 2013 #7

    SteamKing

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    What does this mean?

    The circumference of the whole circle of radius 3 is 6*pi = 18.85
     
  9. Mar 20, 2013 #8
    sorry i'm getting 2.189182969
     
  10. Mar 20, 2013 #9

    Curious3141

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    Correct. ##3\arctan \frac{2}{\sqrt{5}}##
     
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