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I have a problem that goes like this:

"In a circle with radius $6$, what is the measure (in degrees) of an arc whose length is $2\pi$?"

I thought that the answer would be $360$ degrees because $2\pi$ radians is one circle, but the answer in the book is $60$ degrees. The book solved it like this:

$$Circumference=2\pi r=12\pi$$ $2\pi$ is $1/6$ of the circumference: $$1/6\times360=60$$ This route makes sense to me, but I am really confused on the part that it said the length of the arc was $2\pi$, and $2\pi$ radians is $360$ degrees. I drew the conclusion that it meant radians because I was told that you assume that a measure is in radians unless you are told that it is something else. I have just learned about radians, and I obviously made a mistake. Can someone explain to me where I went wrong?

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By length they mean the arc length, not the angle measure.

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    $\begingroup$ Oh! Duh.... Thanks $\endgroup$ – Blake Jun 5 '16 at 20:00
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    $\begingroup$ Yeah it can be slightly confusing because usually when you see $2 \pi$ you automatically think radians, but here this is the arc length. $\endgroup$ – M10687 Jun 5 '16 at 20:00
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Arc arc of $2\pi$ rd has length $2\pi$ on a circle of radius $1$. The length of an arc of $\theta$ rd is equal to $\theta\times\text{radius}$. Whence, if the length is $2\pi$ the arc measure in radians is $\dfrac{2\pi}6=\dfrac\pi3$, i.e. $60$°.

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The length of an arc is equal to $r\theta$ where $r$ is the radius and $\theta$ the measuer of the angle in radians so we have $$r\theta= 2\pi$$ Hence, since $2\pi=360^{\circ}$ it is obvious that the asked angle is equal to $\frac{360}{6}$.

Thus the answer is $60^{\circ}$

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First of all, a radius is a length (not a number). Similarly, an arc (length) is a length (not a number). We have to assume that, in the stated problem, the radius and the arc-length are both measured in the same length units, say LU.

Secondly, the (dimensionally correct) relationship between arc-length, s, radius, r, and central angle, theta, is:

s = r theta/rad

[not "s = r theta"--as is commonly (almost universally!) written.]

where rad is a reference angle of one radian. [The quantity theta/rad is sometimes called the "radian measure of the central angle"--but unfortunately (and confusingly) written as "theta."]

In this case we have: s = 2 pi LU, r = 6 LU. So:

2 pi LU = (6 LU) theta/rad

giving: theta = (2 pi)/6 rad = (2 pi rad)/6. Since 2 pi rad = 360 degrees (= 1 rev), the answer is:

theta = 60 degrees.

--B P Leonard, Emeritus Professor of Mechanical Engineering, The University of Akron.

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