# S=Rθ and Circumference * ratio of arc angle to 360 degrees

• Aug 23rd 2013, 04:53 PM
S=Rθ and Circumference * ratio of arc angle to 360 degrees
I don't know if I'm doing this proof right. But, there are 2 ways to figure out an arc length of a circle s=rθ, and finding the circumference of a circle and multiplying that by the ratio of the angle of the arc/360.

So here is my proof, let's start out with the latter method of finding the arc length.

1. 2*π*r((θ*(180/π))/360) = s
Ok, the part where I take theta and multiply by 180/π is to convert degrees to radians. And then I take that and divide by 360 degrees.
2. 2*π*r(θ*(180/π)*1/360) = s
3. 2*π*r(θ*(1/2π)) = s
4. 2*π*r(θ/2π) = s
5. 2*π*r*θ/2π = s
6. r*θ = s

So, that's my proof. Is there something wrong with the proof?

Circumference * ratio of angle arc/360
• Aug 23rd 2013, 05:36 PM
Plato
Re: S=Rθ and Circumference * ratio of arc angle to 360 degrees
Quote:

Originally Posted by roadran422
1. 2*π*r((θ*(180/π))/360) = sto followi
Ok, the part where I take theta and multiply by 180/π is to convert degrees to radians. And then I take that and divide by 360 degrees.
2. 2*π*r(θ*(180/π)*1/360) = s
3. 2*π*r(θ*(1/2π)) = s
4. 2*π*r(θ/2π) = s
5. 2*π*r*θ/2π = s
6. r*θ = s

I admit not to follow what you are doing.
So let's look at the actual definition of radian.

One radian is defined as the measure of the central angle in any circle that subtends an arc of length equal to the radius of that circle.

Most of us in prefer that we all use radian measure and not use degrees.
That is not a popular position in the mathematics education community.
But among professional mathematicians it is the universal option.
• Aug 23rd 2013, 05:40 PM
Re: S=Rθ and Circumference * ratio of arc angle to 360 degrees
I think I'll retry my question with, where did s=rθ come from?
• Aug 23rd 2013, 06:00 PM
Plato
Re: S=Rθ and Circumference * ratio of arc angle to 360 degrees
Quote:

Originally Posted by roadran422
I think I'll retry my question with, where did s=rθ come from?

But that is exactly my point. I posted the answer to your question.
What don't you understand about that?
• Aug 23rd 2013, 10:02 PM
Prove It
Re: S=Rθ and Circumference * ratio of arc angle to 360 degrees
If you have a circular arc and the angle is measured in radians, your angle tells us the proportion of the circumference (i.e. the proportion of \displaystyle \begin{align*} 2\pi \end{align*} radians) that your arc covers.

So that means \displaystyle \begin{align*} s = \frac{\theta}{2\,\pi} \cdot 2\,\pi\,r = \theta\,r \end{align*}.