Degrees Suck!

[DrunkenArse]

It seems to be common, at least in my country, to measure angles in a unit known as "the degree". This is unfortunate as the degree in truly a primitive and barbaric unit of angular measure. Supposedly, its origins lie in the fact that ancient cultures divided the year into 360 days so the degree divides the circle into 360 segments. This is quite a poor way to do mathematics. There is a solution! The Radian is a natural unit of angular measure. However, it is not often introduced until the student's mind has already been corrupted by exposure to degrees. This is highly unfortunate in that it encourages the student to think in terms of degrees when trying to understand radians. The fundamental problem that both units are trying to address is providing a way of measuring how "large" angles are. Suppose we have an angle: How are we to determine how large this angle is? Both methods begin with superimposing a circle over the angle: Doing so provides a beautiful way to answer our question which the method of degrees completely overlooks. We have the equation: c = 2πr This says that the circumference of a circle is equal to two times pi times the radius of the circle. Let us normalize the circle so that it has radius 1. We can do this by dividing both sides by the radius. c/r = 2π We see that a circle of radius 1 has a circumference of 2π. The idea of radians is to declare that we will measure angles by the length of the portion of the circle which is contained within the angle after the circle is normalized to radius 1. So a full circular angle (that is turning around in a full circle) is 2π radians. A semi-circular angle is π radians and a quarter-circular angle is π/2 radians. In general, if an angle goes around a circle x times, then it is 2πx radians. So an angle that goes around a circle 1/8 times is π/4 radians. Another indication of the elegance of radians is in the simplicity of measuring them. Whereas one needs to carry around a circle with 360 marks around it to measure degrees, one needs only a length of string to measure a radian. Simply lay the string along the portion of the edge of the circle that is contained within the angle. Then see how many times you can mark off the radius of the circle along the same length of string. That is the "size" of the angle in radians. Granted, one could measure the degrees in this manner too but only by measuring it in radians and then converting to degrees. Finally, suppose you want to know how long a circular arc is for some angle. If the angle is specified in radians, you may simply multiply the angle times the length of the radius. arclength = αr We see that we recover the formula for the circumference of the circle when α = 2π.