Angles, arc lengths, and trig functions — Basic example

in the figure at left I put pasted it up here 0.0 0.0 is the center of a circle of radius one point five we see that right over there the missing sector of the circle has central Aang had this the missing sector of the circle has central angle AOB equal to 2 PI over 3 radians that's this right over here and that's the central angle for the missing for the missing sector right over there what is the length of Arc AC B now ACB ACB is this arc it's kind of the rest of the circle the part that's not missing so this is ACB right over here well the way that I would tackle this I would think well what's the circumference of the entire circle and then what fraction of the entire circle is this arc length so let's first think about the circumference of the entire circle circumference is equal to 2 pi times R in this case our our our radius is 1.5 this is going to be 2 pi times 1.5 now 2 times 1.5 is 3 and so this is going to be equal to this is going to be equal to 3 PI so the circumference of the entire circle is 3 pi and now there's a couple of ways you could do it you could figure out well what is the length of this arc right over here and then subtract that from the circumference and then you'd be left with the magenta part or we could figure out the central angle of the magenta part we could figure out this angle and think about well what fraction is that going to be is this of the if we were to go all the way around and if we're thinking in radians going all the way around is 2 pi radians so what fraction is this angle of 2 pi and then that's going to be the same fraction that this arc length is of the entire circumference well what's this angle going to be well it's going to be if we won't remember if we went all the way around if we went all the way around the circle if we went all the way around the circle that would be 2 pi radians but if we want to figure out this magenta central angle it's going to be 2 pi minus this 2 pi over 3 so this is going to be two pi let me do it in that magenta color so the central angle for this piece of the circle for that's kind of the central angle for ACB is going to be two pi minus 2 PI over 3 I'm going all the way around but then I'm subtracting out this part right over here now what's 2 pi minus 2 PI over 3 let's say I can find a common denominator instead of writing it as 2 pi I can write that as 6 PI over 3 so let me do that it's gonna be 6 PI over 3 minus 2 PI over 3 well that's going to be 4 PI over 3 4 PI over 3 so once again this angle right over here is 4 PI over 3 for 4 PI over 3 radians now what fraction is that of if we were to go all the way around the circle well once again this central angle is 4 PI over 3 if you were to go all the way around the circle that's 2 pi so this is the fraction of the entire circle that this arc represents and so let's just multiply that times the entire circumference times 3 pi now let's try to simplify it let's see we have pi divided by PI and see if we take this 3 and multiply it times the numerator this 3 is going to cancel with that 3 and we're gonna left would be left with 4 PI over 2 4 pi divided by 2 is equal to is equal to 2 pi so that's the length that's the length of this arc so it's it's actually exactly 2/3 of the entire two-thirds of the entire circumference so let me just select that 2 pi and we're done