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

You want to use C = 2Pi (r,) where r is radius and C is the circumference. This has two unknowns of C and r, which we want the latter. But C = 3 (6) = 18, because if an arc length of 6 is subtended by 120 degrees, then and arc length of C = 18 is subtended by 360 degrees,

Now we solve for r getting 18 / 2Pi.

Answer 2

fraction of the circumference = 120/360 = 12/36 = 1/3

So, 6 = 1/3 of the whole circumference

Written as an equation...

6 = 1/3 * C

18 = C

C = 2*pi*r

r = C/(2*pi)

r = 18/(2*pi)

r = 9/pi

r = radius = 9/pi ~ 2.87

Answer 3

Convert the angle to radians, and use the relation

r = s/θ

s = 6

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

Plug in the numbers and get r.

Answer 4

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).