SOLUTION: Find the number of radians in the central angle that subtends an arc of 6m on a circle of diameter 5m.

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Question 1169428: Find the number of radians in the central angle that subtends an arc of 6m on a
circle of diameter 5m.
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
the diameter is 5 meters means the radius is 2.5 meters.
the circumference of the circle is equal to 2 * pi * r.
since r = 2.5, the circumference is equal to 2 * pi * 2.5 = 5 * pi meters.
the length of the arc is 6 meters.
the ratio of the length of the arc to the length of the circumference of the circle is 6 / (5 * pi).
since the number of radians in the central angle of full circle is equal to 2 * pi, then the number of radians of the central angle in the section of the circle with an arc of 6 meters is equal to 6 / (5 * pi) * (2 * pi)
this simplifies to (12 * pi) / (5 * pi) radians which results in 12/5 radians which results in an angle of 2.4 radians.
2.4 radians / 2 * pi radians = .3819718634.
that's the radio of the angle of the arc to the angle of a complete circle.
that is also the ratio of the length of the arc of 6 meters to the circumference of the circle that is equal to 5 * pi.
divide 6 by 5 * pi to get a ratio of .3819718634.
this confirms the angle of the arc is correct, because the ratio of the central angle formed by the arc to the central angle of the circumference of the circle is in proportion to the length of the arc divided by the length of the circumference.

note that, if you want to express an angle in radians into an angle expressed in radians * pi, all you have to do is divide the radians by pi to get radians * pi radians.
example:
2.4 radians equals (2.4 / pi) * pi radians which equals .7639437268 * pi radians.
to convert back to radians, you simply multiply .7639437268 by the value of pi to get 2.4 radians.

here's what looks like a good reference.

https://tutors.com/math-tutors/geometry-help/how-to-find-arc-measure-formula#:~:text=An%20arc%20measure%20is%20an,a%20n%20s%20%3D%20180%20%C2%B0%20.

let me know if you still have questions regarding this.

theo


Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
Find the number of radians in the central angle that subtends an arc of 6m on a
circle of diameter 5m.
This SIMPLE problem has a formula, which is: , where:  is in radians

                                                               is the length of the arc

                                                               is the radius of the circle

 then becomes:  

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