I have been teaching my brother some trignometry. There is a formula as arc length of circumference of a circle. The basic formula is $$l = r\theta.$$ But sometimes for length they use $l = 2r$ and other times $l = 2\pi r$. I want to know when to use $l = 2r$ and when to use $l = 2\pi r$.
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1The formula $l=r\theta$ is right, your other formulae correspond to specific choices of $\theta$. – frog Jan 08 '15 at 09:07
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1(Remember that $\theta$ has to be in radians, by the way.) – Akiva Weinberger Jan 08 '15 at 11:25
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Conventionally, $2r$ refers to the diameter. So you actually want to say $d=2r$, using $d$ for diameter instead of $l$.
The reason why is that $l$ is already used for arc length, and you do not want to create ambiguity with diameter $d$.
Now, $l=2\pi r$ is the circumference of the circle. This is the arc length formula $l=r\theta$, with $\theta=2\pi$.
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If the angle of the arc is $\theta$ in radians, the length of the arc subtended by this angle is, as you correctly wrote, $$r\theta$$
A specific case of the above formula is the total circumference of a circle. This is the same as the arc length corresponding to a full angle of $2\pi$, so the circumference is $$2\pi r$$
You also mentioned $2r$. You probably saw this in an expression for the diameter of a circle (which is precisely twice as long as the radius). A less likely possibility is that you encountered an arc with length $2r$, corresponding to an angle of $2$ radians on a circle with radius $r$.
Zubin Mukerjee
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