# Video: Finding the Length of the Arc given Its Measure in Radians and Its Circleโs Radius

An arc has a measure of 2๐/3 radians and a radius of 9. Work out the length of the arc, giving your answer in terms of ๐, in its simplest form.

02:05

### Video Transcript

An arc has a measure of two ๐ over three radians and a radius of nine. Work out the length of the arc, giving your answer in terms of ๐ in its simplest form.

Okay, to help us understand whatโs going on in this question, Iโve drawn a little sketch. And now Iโm gonna annotate it with the values we know. So first of all, Iโm gonna actually show the angle. And this angle thatโs actually measured in radians and itโs two ๐ over three radians. Thatโs what the little ๐ means. That means radians.

Next, we know that the radius is equal to nine. Okay, great! And now finally, we need to look at what weโre actually looking for. But what weโre wanting to find in this question is actually the length of the arc, OK, so itโs ๐, which weโll be looking for.

Okay, great! So now visualize what weโve got. Now letโs put it into a formula. Well, the formula weโre actually gonna use is the formula for the length of an arc. And we know that the length of an arc ๐ is equal to ๐๐. But one important thing to remember is that ๐ must be in radians. And thatโs going to be key for this.

And actually, as we look back to the question, we know that, for this question, we have been told that ๐๐, which is two ๐ over three, is in radians. So great! We can use our equation. So first of all, what Iโve actually done is Iโve written down what we know. So we know ๐ is equal to nine. ๐ is equal to two ๐ over three. And ๐ is what weโre trying to find out.

So now what weโre going to do is weโre actually going to substitute these values into our equation, which is ๐ is equal to ๐๐. So then we get the equation ๐ is equal to nine multiplied by two ๐ over three. So therefore, we can say that ๐ is equal to 18๐ over three.

So is that us finished? Have we done? Well, if we look at the question, we say, โwell, yes it is in terms of ๐.โ But is it in its simplest form? And no, weโve got one more step to complete because if we want to find out what ๐ is in its simplest form, then weโve gotta do 18๐ divided by three. So 18 over three is six. So therefore, we can say that the length of the arc ๐ is equal to six ๐ in its simplest form.