## anonymous one year ago (will give medal to the person who helps) Circle O has a radius of 5 centimeters and central angle AOB with a measure of 60° Describe in complete sentences how to find the length, in terms of a radian measure, of . arc AB (picture will be below)

1. anonymous

|dw:1440122370445:dw|

2. mathstudent55

$$s = r\theta$$

3. anonymous

well I would like a explanation to ..

4. anonymous

|dw:1440133220991:dw| 180 degree= pi radians $60~ degree=\frac{ \pi }{ 180 } \times 60=\frac{ \pi }{ 3 } radians$ $l= r \theta =5 \times \frac{ \pi }{ 3 }=?$

5. anonymous

wouldnt the fraction be pi ? pi is 3.14 right? and the fraction it pi over 3 ? @surjithayer

6. anonymous

it would make it a whole number right?

7. mathstudent55

The definition of a radian: A central angle of 1 radian tends an arc of length 1 radian in a unit circle.

8. mathstudent55

|dw:1440123188616:dw|

9. mathstudent55

The above is true for a unit circle.

10. anonymous

you're confusing me :( idk where you got s from.... ~~ @surjithayer Was I correct about what I said ?? I got 15.7 for the answer.

11. mathstudent55

|dw:1440123441936:dw|

12. anonymous

in my lesson it didn't say anything about that symbol, and i thought the arc is the same as the center angle.

13. mathstudent55

Here is the general case for any radius. A full circle has circumference $$2\pi r$$. The circumference of a full circle can be thought of as the arc length of an arc whose central angle is $$2 \pi$$ radians. For a full circle, s, the circumference is: $$s = 2 \pi r$$ For any central angle, $$\theta$$, the arc length is a fraction of the full circle's central angle, $$2 \pi$$. $$s = \dfrac{\theta}{2 \pi }\ \cdot 2 \pi r$$ $$s = \dfrac{\theta}{\cancel{2 \pi}}\ \cdot \cancel{2 \pi} r$$ $$s = r \theta$$ This is how you get the formula $$s = r \theta$$ If you have a circle of radius r, and a central angle of $$\theta$$ radians, then the arc length is simply $$s = r \theta$$

14. anonymous

yes thats the equation right ??

15. anonymous

and the other person plugged in the numbers correctly ?

16. mathstudent55

You have a central angle of 60 degrees and a radius of 5 cm. You can use the formula $$s = r \theta$$, but you must convert 60 degrees to radians first. $$60^\circ \times \dfrac{\pi ~radians}{180^\circ} = \dfrac{\pi}{3} ~radians$$ Now that we have the angle of 60 degrees converted to radians, we use the formula $$s = r \theta$$ $$s = 5 ~cm \times \dfrac{\pi}{3}$$ $$s = \dfrac{5 \pi}{3} cm$$ That is the exact arc length. If you want an approximate answer, just use 3.14 for $$\pi$$. $$s = \dfrac{5 \times 3.14}{3} ~cm$$

17. anonymous

so it would be pi basicly ? if you times 5 by pi oits 15.7 then you divide it by 5 its pi again.

18. anonymous

@mathstudent55

19. mathstudent55

pi is not exactly 3. It's a little more than 3. When you multiply 5 by pi and divide by 3, you get a number a little larger than 5. 5 * 3.14 / 3 = 5.23

20. anonymous

well alright, thanks for explaining it to me. ~~~ ! It helped alot