If the central angle of a circle has measure 120° and makes a minor arc with length 6, what is the radius?

Question

If the central angle of a circle has measure 120° and makes a minor arc with length 6, what is the radius?

Answer 1

s = arc length

θ = angle in radians

r = s/θ

s = 6

θ = (120°)(π/180°) radians = ? radians

Plug s and θ, with θ in radians, into the equation for r.

Answer 2

Hi Jessica.

You can do this problem using proportions (that is, setting ratios equal to each other). We know that the full circle is 360 degrees and the angle, part of the way around, is 120 degrees. We also know that the full circumference is 2πr and the part of the circumference subtended by the angle is 6. Remember that these are in proportion, which means that

part way around part of the circumference

------------------------ = -----------------------------------

whole way around whole circumference

Now we substitute

120 degrees 6

------------------ = ------

360 degrees 2πr

...and then solve for r (I'll leave that to you).