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Arc Length

In mathematics, the length of an arc of a curve at a particular point is the measure of the segment of the curve joining the endpoints of the arc. The length of the curve between two points is the length of the shortest curve between them. The length of an arc of a curve at a particular point is the measure of the segment of the curve joining the endpoints of the arc.

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Video Transcript

Hey, guys, we're here to talk about arc length today. That would be what is the actual physical length measurement of part of a circle circumference. So if you're talking about having a circle and let's say a partition that circle with the central angle And I want to know what is the fiscal length of this ark, how do we do that? Well, let's call this angle here theta. No, this is all proportional. Theta is just a chunk of 360 which is the total revolution of an entire circle. Right? We know that the entire revolution in a circle is gonna be 360 theater right now. Is this, um, portion of if I call this length here l for the length of the ark that I can say l is gonna be proportional to the entire circumference. Just like Fada is 260. Recall that your circumference formula is two pi r. So this is how you can find the length of an arc through this proportion. So to solve for this, we're gonna multiply by two pi r. So we're gonna essentially have two pi r times Data over 3 60 equals your length of the Ark. A quick example. I have a circle. I know the angle is 110 and the radius is eight. What is the ink? The length of this art. Pardon me. So you would say All right. 1 10 is 23 60. So the part is to the whole, as the part is to the whole. Now the circumference is going to be two times pi times eight. Or in other words, in other words, 16 pie. So if I multiplied by 16 pie, we're gonna get 16 pie times 1 10/3. 60. Now, let's do a little reducing. Yeah. 16. Divided by 3. 60. Don't. That's two 45th. So you're gonna get this part here is to 45th. So we're going to get 220. Were this a little nicer? We're gonna end up with 220. Hi. Over 45 to 20. Divided by 45. We can reduce even further. 2 44 pi over nine. Or in other words, 44 pi over nine. Okay. Gets you roughly 15.4 units. So the key here was finding or knowing the proportion that you need, which is part the whole is part to hold. Okay, so that's how you're gonna find arc length.