Calculating Arc Length of a Circle: What's the Correct Formula?

[icesalmon]

Homework Statement

Find the Arc Length from (0,3) clockwise to (2,sqrt(5)) along the circle defined by x² + y² = 9

Homework Equations

Arc Length formula for integrals

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.

 

[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 $s=r\theta$

 

[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°.

 

[SammyS]

Homework Statement

Find the Arc Length from (0,3) clockwise to (2,sqrt(5)) along the circle defined by x² + y² = 9

Homework Equations

Arc Length formula for integrals

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.

If you want the arc length geometrically (perhaps trigonometrically would be better terminology) then just use

Arc length = R(θ₂-θ₁) ,

where θ₂ = π/2

and θ₁ = arccos(2/3)​

 

[icesalmon]

I'm getting 2 and some change.
thanks.

 

[SteamKing]

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

 

[SteamKing]

I'm getting 125 and some change.
thanks.

What does this mean?

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

 

[icesalmon]

sorry I'm getting 2.189182969

 

[Curious3141]

sorry I'm getting 2.189182969

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