what exactly is a Radian have you ever wondered the answer to that question what is a Radian what is it well in this video we're gonna talk about that now a Radian is a unit of angle measure let me say that again a Radian is a unit of angle measure one Radian is equal to an angle measure of approximately 57 point 3 degrees so that's really what it is now how do we get that number how do we get that one Radian is equal to approximately 57 point three degrees well let's draw a circle how many degrees is one complete circle so if we travel starting from one point on a circle all the way to another two back to that same point if we make one full rotation what is the angle in degrees that we have traveled a full circle is equal to 360 degrees and it's also equal to two pi radians so 360 degrees is equal to two pi radians so to get the angle measure of one Radian in degrees take 360 degrees and divided by two pi radians now keep in mind PI even though you could use the rounded value of 3.14 it's equal to 3.14159265 for so using that number in this equation you may want to put this in parenthesis by the way if you type in into your calculator 360 divided by 2 pi you're gonna get this value fifty-seven point two nine five seven seven nine five one which if you round it it's approximately 57 point three degrees so that is a Radian it's an angle measure it's a unit of angle now that we know what a Radian is how did we get that number in the first place where does it come from well let's talk about that first let's draw an imperfect circle so that wasn't that bad for my first try and let's draw the radius of the circle the radius of the circle is the distance between the center of the circle which we're going to make it the point highlighted in red is the distance between the center of the circle and any point on a circle so that's the radius of the circle now we're going to draw another radius the distance between these two points let's call this point a and point B that is known as the intercepted arc or you can call it the arc life which most textbooks they would use the symbol s for that the arc length is equal to the angle measure times the radius of the circle so this is the angle measure now the arc length could be anything from 0 to 2pi even more than 2pi but when the arc length is equal to the radius of the circle the angle is equal to 1 a Radian and that's where the idea of a Radian comes from one Radian occurs when the arc length is equal to the radius of the circle so that's where we get it and you could draw any circle if you take the measurement of the radius of the circle and you draw these two in such a way that the arc length is equal to the radius of the circle you're gonna get an angle of approximately 57 point three degrees or one Radian so here's how you can use this formula to get that answer we're gonna make s equal to R so I'm gonna be place s with R if you divide both sides by r r divided by r is 1 and so when s is equal to R theta is equal to 1 more specifically one Radian now you might be wondering what are the units of s theta and R so for instance let's say if we have a circle with a radius of 4 inches now let's say the angle measure is 2 radians what is the arc length what is s equal to now using the formula you would say that beta is 2 R is 4 so s would be 8 but specifically 8 inches let's make sense of the units theta is in radians so it's 2 radians over 1 in order for us to get inches for s the radius have to be 4 inches per Radian so that these two units will cancel and s will be 8 inches so the radius is really a unit of left per angle measure in this case it's 4 inches per Radian now let's answer another question that relates to radians how many radians are in a circle what would you say how many radians are in one circle so let's draw a circle the circumference of a circle is basically the distance around a circle technically this distance highlighted in red so that is the circumference of a circle the circumference of a circle is equal to 2 pi times R now this equation comes from the arc life equation s is equal to theta times R so let's say this is the radius and this is the radius s will be equal to the intercepted arc however we could extend us such that equals the entire circumference of a circle so when s is equal to the circumference of the circle R is still equal to R but that means that the intercepted angle is equal to 2 pi radians so for one complete circle the angle measure of that is 2 pi radians or 360 degrees now we know that pi is approximately 3.14 I mean you could use some other numbers like 3.14159 if you want to but 3.14 will work for our purposes if we multiply that by 2 then we get that the angle measure is approximately six point two eight three radians if you round it so this is how many radians there are in a complete circle there are six point two eight three radians in one circle there's one more thing that we need to talk about related to this topic and that is the ability to convert from radians to degrees and degrees to radians so let's say if you have an angle measure of 30 degrees how do you convert that to radians feel free to pause the video if you want to try this so to convert degrees into radians multiply the angle in degrees by PI divided by 180 keep in mind we said that 360 degrees is equal to PI radians I mean 2 pi radians let me take that back so if we divide both sides by 2 360 divided by 2 is 180 2 pi divided by 2 is PI so 180 degrees is equal to an angle measure of pi radians thus to convert 30 degrees into radians multiplied by PI over 180 degrees we want the unit degrees to cancel so we're going to get 30 PI over 180 now what we need to do is simplify this fraction the first thing we can do is cancel a 0 so we get 3 PI over 18 and then 18 we can break that up into 3 times 6 3 times 6 is 18 so this is going to give us PI over 6 radians let's try another example go ahead and convert 60 degrees to radians so let's start with what we're given 60 degrees now we're going to multiply that by PI over 180 degrees so these units will cancel we're gonna get 60 PI divided by 180 and then we can cancel a 0 which gives us 6 PI over 18 and 18 let's rewrite that as 6 times 3 so we can cancel a 6 and thus we're going to get PI over 3 radians so 60 degrees is equal to PI over 3 now what about converting in the other direction that is let's say if we're given an angle in radians and we want to convert it to degrees how can we do that so let's start with an angle measure of 5 PI over 6 radians in order to convert it to degrees you want to flip the second fraction instead of multiplying it by PI over 180 you want to multiply it by 180 degrees over PI radians so the unit PI will cancel thus we're going to get 5 times 180 divided by 6 now 180 is basically 30 times 6 let's keep the degree symbol here so we could cancel a 6 and then we have 5 times 30 5 times 3 is 15 if we add the 0 this is going to be 150 and so that's how we can convert from radians to degrees so 5 PI over 6 radians is equal to 150° let's try another example let's say we have the angle 4 PI over 3 let's convert that to degrees so just like before we're going to multiply by 180 degrees over PI so the unit PI will cancel and this is going to be 4 times 180 divided by 3 180 is equal to 60 times 3 so canceling 3 we have 4 times 60 left over 4 times 6 is 24 so 4 times 60 is 240 thus 4 PI over 3 is equal to 240 degrees so that's it for this video so now you know what radians are where it comes from and you know how to convert between radians and degrees and vice versa thanks for watching