An arc has a measure of two 𝜋 over three radians and a radius of nine. Work out the length of the arc, giving your answer in terms of 𝜋 in its simplest form.
Okay, to help us understand what’s going on in this question, I’ve drawn a little sketch. And now I’m gonna annotate it with the values we know. So first of all, I’m gonna actually show the angle. And this angle that’s actually measured in radians and it’s two 𝜋 over three radians. That’s what the little 𝑐 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 we’re actually looking for. But what we’re wanting to find in this question is actually the length of the arc, OK, so it’s 𝑙, which we’ll be looking for.
Okay, great! So now visualize what we’ve got. Now let’s put it into a formula. Well, the formula we’re actually gonna use is the formula for the length of an arc. And we know that the length of an arc 𝑙 is equal to 𝑟𝜃. But one important thing to remember is that 𝜃 must be in radians. And that’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 𝑟𝜃, which is two 𝜋 over three, is in radians. So great! We can use our equation. So first of all, what I’ve actually done is I’ve written down what we know. So we know 𝑟 is equal to nine. 𝜃 is equal to two 𝜋 over three. And 𝑙 is what we’re trying to find out.
So now what we’re going to do is we’re actually going to substitute these values into our equation, which is 𝑙 is equal to 𝑟𝜃. So then we get the equation 𝑙 is equal to nine multiplied by two 𝜋 over three. So therefore, we can say that 𝑙 is equal to 18𝜋 over three.
So is that us finished? Have we done? Well, if we look at the question, we say, “well, yes it is in terms of 𝜋.” But is it in its simplest form? And no, we’ve got one more step to complete because if we want to find out what 𝑙 is in its simplest form, then we’ve gotta do 18𝜋 divided by three. So 18 over three is six. So therefore, we can say that the length of the arc 𝑙 is equal to six 𝜋 in its simplest form.