# Getting the correct arc length formula

You have your regular arc length formula: $$l = r\theta$$ where $$\theta$$ is in radians. Now you want to express the $$\theta$$ in degrees.

So you apply the $$2\pi [rad] = 360[degrees]$$ formula to the above equation, and you get $$1[rad] = \frac{360 [degrees]}{2\pi}$$ $$l=r\theta_{rad}=r\theta \frac{360degrees}{2\pi}$$ But my book says when you want to get arc length $$l$$ in degrees, the correct formula is $$2\pi r\frac{\theta}{360}$$, which is different from what I got above.

Can someone please explain why I get this result? Also apologies if this question is too simple to be posted here. I searched online but couldn't find a proper resource. Many thanks in advance!

[Update] I missed a really simple definition which was $$\theta_{deg} = \theta_{rad} * \frac{360}{2\pi}$$. After realizing it, everything began to make sense. Thanks to everyone that shared their answers!

$$360^\circ = ~$$ an arc length of $$~2\pi.$$

Therefore, $$~1^\circ = ~$$ an arc length of $$\displaystyle ~\frac{2\pi}{360}.$$

Therefore, $$~l^\circ = ~$$ an arc length of $$\displaystyle ~\left[l \times \frac{2\pi}{360}\right].$$

Another way of saying the same thing is that

$$~l^\circ = ~$$ corresponds to $$\displaystyle ~~\left[l \times \frac{2\pi}{360}\right]~$$ radians.

We can work this backward from the original formula $$l=2\pi r \frac{\theta[degrees]}{360}$$ $$l=r(\theta[degrees]\frac{\pi}{180})$$ We know that to convert $$\theta$$ in degrees to radians, we need to multiply by $$\frac{\pi}{180}$$. We use this in the equation and get $$l=r\theta[radians]$$ which is the regular arc length formula.

To convert $$\ell = r\theta$$ where $$\theta$$ is in radians to degrees, remember that $$\theta = \frac{\pi}{180^\circ}\varphi = \frac{2\pi}{360^\circ}\varphi$$ where $$\varphi$$ is in degrees. Hence, $$\ell = r\left(\frac{2\pi}{360^\circ}\varphi\right) \\ \ell = 2\pi r\left(\frac{\varphi}{360^\circ}\right).$$ Then, just rename $$\varphi$$ to $$\theta$$ if you like.