Photo Credit: Chris Discenza
The number of degrees in a circle was historically chosen to be 360 degrees by the Babylonians because it is easily divisible by many numbers without a calculator. Calculators were very expensive back then.
A better angle measurement would be self contained. This means that it wouldn’t depend on an arbitrary number like 360. It would only depend on the geometry. This angle measurement is called the radian. To construct a radian we will consider the definition of an angle. An angle is formed by two intersecting rays. The angle is the fraction of a circle between the rays that resembles a piece of pie. There are actually two pieces: a small piece and the rest of the pie. We will modestly choose the smaller piece. We know that the circumference of a circle is 2R. Since our piece of pie is a fraction of the whole, the length of the crust is a fraction of 2 R. Let’s say that our angle is ¼ of a circle. Then the length of the crust will be ¼ of 2R which is ½R. Now since an angle doesn’t depend on the size of the rays or the circle, it will certainly work for R=1 which is a unit circle. This means that the size of our pie piece is ½ pi. Thus our angle measure is a fraction of pi! We say that our angle is ½ radians. Notice that ½ radians is equivalent to a right angle of 90 degrees. And ½ is a quarter of 2 just as 90 is a quarter of 360. Now see if you can find out how many radian and degrees a third of the pie is.