Video: Finding the Length of the Arc given Its Measure in Radians and Its Circleโ€™s Radius

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

02:05

Video Transcript

An arc has a measure of two ๐œ‹ over three radians and a radius of nine. Work out the length of the arc, giving your answer in terms of ๐œ‹ 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œ‹ 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‘Ÿ๐œƒ. But one important thing to remember is that ๐œƒ 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‘Ÿ๐œƒ, which is two ๐œ‹ 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‘Ÿ is equal to nine. ๐œƒ is equal to two ๐œ‹ 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‘Ÿ๐œƒ. So then we get the equation ๐‘™ is equal to nine multiplied by two ๐œ‹ over three. So therefore, we can say that ๐‘™ is equal to 18๐œ‹ over three.

So is that us finished? Have we done? Well, if we look at the question, we say, โ€œwell, yes it is in terms of ๐œ‹.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œ‹ 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œ‹ in its simplest form.

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