Question

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Find the length of the radius for circle C.

Answer 1

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

Answer 2

C= 2 pi r

Answer 3

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

Answer 4

Now look in your figure.

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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

Answer 11

Let me explain more clearly.

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

Ok so

Answer 17

What would the circumference be

Answer 18

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.

Answer 19

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

Answer 20

\(s = r \theta\)

\(r = \dfrac{s}{\theta} \)

You have s, the arc length, and theta, the central angle. Plug them is and find r.

Answer 21

I got it thanks so much

Answer 22

I have one more question.

Answer 23

have you been paying attention or not?

now you are being asked to solve for s

the arc length, \(s = \large r \times \theta \)

Answer 24

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.

Answer 25

Last question.

Answer 26

You mean the red section of the circle?