I looked all over the internet for this answer, but couldn't find it

[flyingpig]

Homework Statement

Can someone actually explain, without using Calculus, why the arclength formula is s = r*theta

The Attempt at a Solution

I tried doing this in polar coordinates, but I couldn't think about how to do it.

 

[SammyS]

What's the arc length if θ = 2π ? (a complete circle)

 

[armolinasf]

I would imagine that it comes from the fact that a circle has a circumfrence of 2pi*r where theta is just 2pi.

Since the radius of a circle is just a special type of arc. This would only work if the arc has a constant radius, but if that changes you have to do summations of of arcs with constant radius. This would eventually lead to the calculus version of making theta into the infinitely small dtheta

 

[flyingpig]

What's the arc length if θ = 2π ? (a complete circle)

2pi*r, where r is the radius, I think I know how you are going to teach it to me, but I am looking for irregular thetas

 

[Dick]

Oh, come on. Once you've ruled out calculus to do the computation you just back to basic geometry. If the angle is 2*pi (a whole circle) then the length is 2*pi*r. That's the definition of 'pi'. If the angle is 2*pi*x then the length is 2*pi*x*r. This is really just definition of 'angle'. You know this, right? What kind of 'proof' are you looking for?

 

[flyingpig]

Can we prove this in polar coordinates?

 

[SammyS]

What do you mean by prove, and exactly what is it you want to prove?

 

[flyingpig]

Let's say I draw an arbituary arc of some angle that's not easy to compute and I want to know the arclength of that arc, why is it s = r*theta. The circumference thing gives me intuition why, but doesn't give me the math that proves to me it is.

 

[SammyS]

s = r*θ works for an arc with constant radius, subtending an angle of θ, but that's not very general or arbitrary.

 

[flyingpig]

But it isn't obvious to me

 

[SammyS]

What is the definition of the measure of an angle in radians?

 

[flyingpig]

pi/180 * angle

 

[SammyS]

No, that's how to convert an angle measured in degrees to radian measure.

 

[SammyS]

The http://en.wikipedia.org/wiki/Radian#Definition" has this in the definition section:

"... More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s /r, where θ is the subtended angle in radians, s is arc length, and r is radius. ..."

I hope that helps.

 

[QuarkCharmer]

You can prove this with a piece of string, a ruler and any circular object using algebra, knowing only that there is some vague relation between the radius of a circle and the radian.