## anonymous one year ago https://media.education2020.com/evresources/2092539_2f57b746-c73f-4a4b-8c7e-ce9ee207b927.png Find the length of the radius for circle C.

1. mathstudent55

When the radius of a circle is r, what is the circumference of the circle?

2. anonymous

C= 2 pi r

3. mathstudent55

Great. The circumference of a circle is an arc that corresponds to a central angle of 360 degrees or 2 pi radians.

4. mathstudent55

5. mathstudent55

You have a central angle of 4pi/9 radians corresponding to an arc length of 20pi/3 units.

6. mathstudent55

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7. nincompoop

maybe explain what $$\pi$$ corresponds to as in a unit circle

8. mathstudent55

If we divide the circumference by the central angle, we get the radius.

9. anonymous

So how do we get the circumference from what we have now

10. anonymous

Since we don't have the radius I'm confused 😕

11. mathstudent55

In a circle, the central angle is $$2 \pi$$ rad. Dividing the circumference by the central angle gives us the radius. $$\dfrac{2 \pi r}{2 \pi} =r$$

12. mathstudent55

Let me explain more clearly.

13. mathstudent55

In this problem we are dealing with an arc length, and a radius, and a central angle.

14. anonymous

But how can we use r when we don't have the radius

15. nincompoop

we can skip the circumference since we are only dealing with an arc the length of an arc is calculated by the formula, $$\large s = r \times \theta$$ you are given the measurements of s and angle

16. mathstudent55

When the central angle is given is radians, the length of the arc is the central angle multiplied by the radius.

17. anonymous

Ok so

18. anonymous

What would the circumference be

19. mathstudent55

Since you are given the length of the arc and the central angle, and we are dealing with radian measures, divide the length of the arc by the central angle and you get the radius.

20. nincompoop

then we know how to approach how to calculate r $$\large \ r = \frac{s}{\theta} \rightarrow \huge \frac{20 \pi}{3} \div \frac{4 \pi}{9}$$ follow the rules when dividing two fractions

21. mathstudent55

$$s = r \theta$$ $$r = \dfrac{s}{\theta}$$ You have s, the arc length, and theta, the central angle. Plug them is and find r.

22. anonymous

I got it thanks so much

23. anonymous

I have one more question.

24. nincompoop

have you been paying attention or not? now you are being asked to solve for s the arc length, $$s = \large r \times \theta$$

25. mathstudent55

In the second problem you use the arc length formula directly. $$s = r \theta$$ You are given r, the radius, and theta, the central angle. just multiply them together.

26. anonymous

Last question.

27. mathstudent55

You mean the red section of the circle?