# Radians VS Degrees & how to get the arc length of a circle

**A free video tutorial from**Mark Misin

Aerospace & Robotics Engineer

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INTUITION MATTERS! - Applied Calculus for Engineers-CompleteCalculus + Engineering + PID: Functions, Limits, Derivatives, Vectors, differential equations, integrals: BEST CALCULUS

34:53:37 of on-demand video • Updated October 2020

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And now I want to draw your attention to something we know that in order to calculate the circumference of the circle we had this formula to pi times are two pi times are. But look essentially it's a straight line function right. Because if Phi draw this X is here and I call R and here circumference is C then well I have a straight line function here. And do you remember you can get the change to see overbilled to our would be to pi. So what does that mean. It means that if the radius of the circle increases by one then the circumference of the circle will increase by 2 pi. So it will be 2 pi meters more. So for instance let's say that my first circle was with a radius of five meters. In my first conference with B to PI 5 which is 10 PI meters. But then if y increase the radius by by 1 metre Let's say now six meters then it would be C to 2 pi. Times 6 would be 12 PI meters and you can see the difference. See two minus see one would be to PI right. So it would be two pi meters. OK. So that's one way to look at it. So we have a fixed circle and you know we have a circle and if we varied the radius of the circle then the circumference of the circle will change. How about How about if I'm not interested in it then what. What if my circle is fixed. The radius is fixed. And what I want is is to calculate the length of the section of circles conference. In other words I want to calculate some kind of arc. Let's see if I take 90 degrees here. I want to know what is the length of this arc. Well in order to do that and let's denote that with length No. Well in this case if that's nine degrees then one way to do it would be 90 over 360 which is 1 4 times 2 conference. So you would have one over 4 times circumference or in-conference divided by 4. Right. But now I'm going to show you another alternative to look at it and I'm going to introduce something called radiance which is extremely important for us. So it's very important it's very important to pay attention to that now. Look what I can do now is that I can keep our fixed and I can say in this formula I can say now I keep our fixed and I let two pi remain two pious feet. So if I very fita then I will get length because the 5 0 seater then I have zero length. I get arclength. So the function here would be length equals R which is now fixed. I'm just going to put f here which is fixed let's say 5 meters and 6 times sæter. So it's it's an on a function now I very very this part of the equation and of course if I get if I get to pi then the length equals the circumference of the circle then if I have pi then it's half of the circle. And if I have 90 degrees which is PI over to then I have one fourth of the circle. And now this angle here. And of course if I want to make more let's say more circles around the circle. If I want to make more than one rotation with two rotations then I will have here later on for pi. And in fact in fact fetor can be very nuts from 0 2 to infinity. But from mine is infinity plus infinity because I can take this radius and I can turn it in this way as well. So the convention is positive rotation is counterclockwise. So you go like this you think of a clock and you go like this. But the negative rotation is clockwise. It's like this. So as you rotate clockwise on this axis you move to the left. And if you rotate counterclockwise like this then you move with to the right on this axis you move to the right. But of course physically physically you cannot have negative lengths. So we'll just we'll just start measuring length something from the deck with zero. We go in this direction. So we. So we. Well in fact you could have this absolute value function where you have length like this. And then as you as you rotate clockwise then your length will still be mirrored mirrored like this line over this length axis OK. But now this is an angle but it's not the grease. It can be because pi is six point to something six point two and it's better because pi itself was three point fourteen. Three point one. So then two pi would be six point six point two eight. Right. And if we had degrees then we would have something like circumference equals 360 times radius. But that's not the case. So for one circle for one every time we complete the circle it's two pi times radius do PI six point to wait. Now it's just another type of angles and we call it radiance. So Radiance is just another way to measure angles for a circle and they are very useful because now if we want to calculate the arc length of our circle then we just take l equals and we take the radius the fixed radius of circle whatever circle we have. And we multiply it by that fetor. And then for example if you want 30 degrees then that would be fetor would be PI over 6 radians. So that would be equivalent to 30 30 degrees. And then you would just multiply the length of 30 degrees and less safe radius this side of Meersch will be five times pi over six meters. And how to get a relationship between radians and degrees. Will we have this function here. We have. L equals nine. Over 90 over 360 degrees right time C but C is to pi are right. So we can multiply just like this to PAJA our OK. But then we can make this the degrees. This is the degree said we're interested in and we have just found out that the length also equals radius times the radiance. Now if this and this are equal which they are then I can write it down like this are times saeter equals 90 over three hundred and sixty times to pi are. So what I can do now I can cancel the radio. One radius and both sides I can cancel them out because they're equal and I can take this guy here and I can just make it as a variable and let it. Just like that very fetor. And there you go. I have this equation see that equals to pi. Over three hundred and sixty times degrees or fita equals PI over 100 and 80. Because this is one and this will be one hundred and eighty times degrees. And as you can see it's also another function and also a straight line function. So I have radians here and I have degrees here and the relationship between them. Is PI over one hundred and eighty. It means that's the change of radians with respect to 1 unit change of degree is PI over one hundred and eighty.