# Thread: s= r x theta

1. ## s= r x theta

Ok, i am confusing myself over something that i thought was really basic and i understood!
Is the arc length formula, s= r*theta , for theta in radians a derived result or a definition?
And is there a very clear way of seeing what it is? Does it come from a definition of 1 radian? If so, how does it develop into the arc length formula!

2. ## Re: s= r x theta Originally Posted by rodders Ok, i am confusing myself over something that i thought was really basic and i understood!
Is the arc length formula, s= r*theta , for theta in radians a derived result or a definition?
And is there a very clear way of seeing what it is? Does it come from a definition of 1 radian? If so, how does it develop into the arc length formula!
The definition of 1 radian is the measure of the central angle that subtends an arc of length equal to the radius.

3. ## Re: s= r x theta

The circumference of a circle with radius $r$ is $2\pi r$

there are $2\pi$ radians in a circle.

Thus there is an arc length of $\dfrac{\theta}{2\pi} \cdot (2 \pi r) = r \theta$

The important definition is that there are $2\pi$ radians in a circle.

So you would say that 1 radian corresponds to the angular measure where the subtended arc length equals the radius.

4. ## Re: s= r x theta

Ah ok..

I guess S = rxtheta is equivalent to C= rx 2pi ? so when theta=2pi , S=C
Is a correct way of looking at it too?

5. ## Re: s= r x theta

Yes, it is. For the entire circle, the "arc-length" is just the circumference and the angle, in radians, is $2\pi$. "$S= r\theta$" becomes "$C= 2\pi r$"