# Arc Length of a Circle

1. Mar 19, 2013

### icesalmon

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. Mar 19, 2013

### Simon Bridge

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 $s=r\theta$

3. Mar 19, 2013

### icesalmon

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°.

4. Mar 19, 2013

### SammyS

Staff Emeritus
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)​

5. Mar 19, 2013

### icesalmon

I'm getting 2 and some change.
thanks.

Last edited: Mar 20, 2013
6. Mar 19, 2013

### SteamKing

Staff Emeritus
And make sure you use radian measure. Degrees won't work.

7. Mar 19, 2013

### SteamKing

Staff Emeritus
What does this mean?

The circumference of the whole circle of radius 3 is 6*pi = 18.85

8. Mar 20, 2013

### icesalmon

sorry i'm getting 2.189182969

9. Mar 20, 2013

### Curious3141

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