 So it turns out that arc length can be used to approximate the third side of an assassin's triangle in the following way. Consider we have two points A and B that live inside the plane and suppose we can take a third point, we'll call it that third point C, such that the distance between A and B from C is really, really, really, really, really, really, really, really, really far away. That is C is relatively speaking far away. The radius of this circle is huge compared to the distance between A and B right here, okay? And so if R is significantly larger than this distance, then you'll notice that the arc length between A and B and the line segment between A and B will be relatively the same size. We can approximate them to be about the same and so if this radius is huge compared to this distance right here, this also means that the angle measure is gonna be really, really, really, really small in comparison, right? So if you have a small theta, then the line segment and the arc length will have approximately the same length. And so we can use that to approximate the distance, the linear distance there using this circular distance. And why would we do that? Well, it turns out the arc length formula is much simpler to use in comparison to the trigonometry one would have to use, otherwise to measure this distance. So let me give you an example of such a thing. So imagine you have a person who's standing here on good old planet Earth and above this person is flying a jumbo jet. And so you get something like this, which I know this is like the best picture you've ever seen of a jumbo jet, but that's our picture right here. And so, you know, this person looking above at the airplane is wondering how far above the ground. What's the altitude of this airplane? Well, if the person's able to measure the angle from the tip to the tail of this airplane here, so he measures the angle subtended by the airplane. And so that's gonna be measured to be something super, super small. This is gonna be 0.45 degrees. So it's not even half of one degree. So very, very small angle. Also, because, you know, this person knows the length of the airplane. The airplane itself is 230 feet. Sort of curious why someone has factual knowledge on the length of a jumbo jet but doesn't know the altitude. But that's beside the point. This is just a trigonometric exercise here. What would then be the radius of this circle where we can then see the person as the center of the circle? That's what we want to find here. So the airplane itself is straight. You know, the distance from the nose to the tail there, that's a straight line, not a circle. But if we pretend it's an arc line, because it's approximately the same thing, we can then use the formula S equals R theta right here, which we don't know R, but we can solve for it. R is gonna equal S over theta, which S is gonna equal 230 feet there. The angle measurements for arc length does need to be in radians here. So we have to convert them. So if theta equals 0.45 degrees, we're gonna multiply that, of course, by the appropriate factor pi over 180 degrees to cancel out the degree measurements there. So the degrees cancel out, like so. And so we get the angle measure is going to equal 45 pi over 18,000. Where did that come from? It's basically just if you move this over by two decimal places, you gotta move this over by two decimal places as well. So you can use that number if you want 45 pi over 18,000. Of course 45 does divide into 18,000 evenly. You end up with pi over 400, which is still a very, very small angle measure, but it's important to be in radians in order to use this. So you're gonna end up with this pi over 4,000. If you're dividing by a fraction, you can just multiply by its reciprocal. So we get 230 times 400 over pi. 230 times 400 is gonna be 92,000. Give myself a little bit more space here. You end up with 92,000 over pi. This is the exact value. We just want an estimate though. So putting this into our calculator, taking 92,000 divided by pi, you're gonna end up with 29,285 feet, which, you know, 30,000 feet is an appropriate distance for a plane to be flying. So that seems actually like a pretty good estimate. And so this gives us an example how one can use arc length to approximate distance between points. Or in this case, we will use arc length to approximate the altitude of this plane where really we had a triangle, right? It was really in a Saucely's triangle, but we approximated the Saucely's triangle. It's approximately the same thing as the sector of a circle. So we can use that when, you know, we're basically saying that a triangle is almost a pizza slice when the radius is very, very big.