# Circles

#### Davekim

##### New member
I'm having trouble with this question,

"A circle with a circumference of 6π ft. has a central angle that traces out an arc length of 4 ft. What is the radian measure of the central angle?"

Can someone please explain the process on how to do this problem so I can understand how to do it?

#### MarkFL

##### Super Moderator
Staff member
Hello, and welcome to FMH!

The arc-length $$s$$, radius $$r$$ and central angle $$\theta$$ are related as follows:

$$\displaystyle s=r\theta\tag{1}$$

The circumference $$C$$ and radius $$r$$ of a circle are related as follows:

$$\displaystyle C=2\pi r\tag{2}$$

What I would do is solve (1) for $$\theta$$, and then solve (2) for $$r$$, and substitute for $$r$$ into the equation resulting from solving (1) for $$\theta$$. Then you will have $$\theta$$ as a function of $$s$$ and $$C$$, both of which you've been given. What do you get?

#### MarkFL

##### Super Moderator
Staff member

Solving (1) for $$\theta$$ we obtain:

$$\displaystyle \theta=\frac{s}{r}$$

Solving (2) for $$r$$ we obtain:

$$\displaystyle r=\frac{C}{2\pi}$$

Hence:

$$\displaystyle \theta=\frac{s}{\dfrac{C}{2\pi}}=\frac{2\pi s}{C}$$

Plugging in the given data:

$$\displaystyle \theta=\frac{2\pi(4\text{ ft})}{6\pi\text{ ft}}=\frac{4}{3}$$