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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!

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3 Answers 3

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$360^\circ = ~$ 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.

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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.

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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.

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