### 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 whatโs going on in this question, Iโve drawn a little sketch. And now Iโm gonna annotate it with the values we know. So first of all, Iโm gonna actually show the angle. And this angle thatโs actually measured in radians and itโs two ๐ over three radians. Thatโs 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 weโre actually looking for. But what weโre wanting to find in this question is actually the length of the arc, OK, so itโs ๐, which weโll be looking for.

Okay, great! So now visualize what weโve got. Now letโs put it into a formula. Well, the formula weโre 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 ๐๐. But one important thing to remember is that ๐ must be in radians. And thatโs 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 ๐๐, which is two ๐ over three, is in radians. So great! We can use our equation. So first of all, what Iโve actually done is Iโve 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 weโre trying to find out.

So now what weโre going to do is weโre actually going to substitute these values into our equation, which is ๐ is equal to ๐๐. 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, โwell, yes it is in terms of ๐.โ But is it in its simplest form? And no, weโve got one more step to complete because if we want to find out what ๐ is in its simplest form, then weโve 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.