>> It's the radian angles that are related to arc length. So the first thing we need to do here is convert from 240 degrees into some number of radians. So let's go through that. 240 degrees corresponds with 180 in the same way as theta corresponds with pi. Multiply 'em by pi on both sides and we have 245 over 180, reducing a little bit we find theta to be 4pi over 3. Now this is some number of radians. Well now, now that we have the angle in radians and I can emphasize that up here, I can say that this is then 4pi--whoops, I want 4pi over 3 in radians. Now we go back to our fundamental understanding of the relationship between the length of an arch and a radian. That is theta in radians is arc length over radius and I suggest that you write the words out here. It's possible to write this as, well, arc length we usually use the letter S and for radius we use R. But I don't want you to memorize a bunch of letters. I would for you to understand a relationship that the angle opening is found by taking arc length and dividing by the radius. So you get the number of radius unit you see in that arc length and that tells you the number of items here for the radians of the angle. Okay. Now what do we know about this? Well we know that the theta, the angle here in radians is 4pi over 3, we know the radius is 4. So multiplying on both sides by 4, we find the arc length to be 16pi over 3 or approximately 16.76 inches. Let's talk about some problems related to the information that we know to this point. Let me set the stage like this. We know that the distance formula is distance is equal to rate times time and if I divide on both sides by time, I find that the rate is distance over time or we can say that speed is distance divided by time. Now imagine the distance along a circle, that it's an arc length kind of idea for the distance, you see and then--so the speed then would be arc length over time if we're running in a circular kind of path and as a related matter, we can calculate angular speed, the speed at which the angle is generated, you see and that's simply the total angle that we're talking about divided by time. Consider this problem. The circular blade on a saw has a diameter of 7.5 inches and rotates at 24,000 revolutions per minute. A, find the angular speed in radians per second and B, find the speed of the saw teeth in feet per second as they contact the wood being cut. The important information in the problem is that we have a diameter of this blade of 7.5 inches and it's turning at 24,000 RPMs, that's revolutions per minute. If we're gonna calculate angular speed, we need the size of the angle and some unit of time. Now we'd like this unit of time to be seconds because we want our unit to be in radians per second and I think it was the request, so we want the time in seconds. Now our speed, though, of the rotation of the blade is 24,000 revolutions per minute. So we have a time unit problem here as well as everything else is going on. Alright. Now let's think about theta, let's think about that angle, the total angle, you see, for some period of time. Now that angle is generated by--with two ideas. It's 24,000 revolutions per minute but every one of those revolutions is how many radians? Well a complete revolution is 2pi radians. So we have 2pi radians for every one of these 24, 000 revolutions per minute. So the theta we're talking about here, the angle that we're talking about is 24,000 times 2pi, that's the angle and it's being generated in a minute because it's revolutions per minute. So we have, let's see, 24,000 times 2pi then for 1 minute it's 60 seconds. You see, we wanna convert this to 60 seconds. So now when the smoke clears on this, we find this to be 80pi radians per second and that is the angular speed, the angular speed of the saw blade.