Why do radians even exist? Why would you use them instead of degrees?
In: Mathematics
If you stand on the corner of a big pie wedge, and measure the angle in radians, you’ll immediately know the length of the crust is the angle x radius. you don’t have the additional steps of multiplying by pi and dividing by 360.
If you cut the crust off to make the pie wedge a triangle, you can’t do this for the straight edge normally. However, if the angle is small enough, it’s good enough for government work. So if you stand at the top of a narrow triangle with two equal legs, the opposite side is pretty much the angle x leg length if the angle is less than 0.5 radians or so.
Now you’ve got a great way for quickly figuring out how big something is at a distance. If you take a piece of glass at arms length and scratch out angles in radians, you can figure out how tall something is if you know they’re 1000 feet away. Just multiply the angle x 1000 feet and you’ll get their height. You could do the reverse, if you know their heigt, divide it by the angle and you get the distance, and you never had to muck about with pi or 360. Note you can do this with any distance unit, eg feet, meters, inches, whatever, you’ll get your answer in the same units.
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