Students hate radians! When they show up on the SAT math section, kids get nervous, because in school, degrees are introduced first as a way to measure angles, and students find comfort in that familiar unit even though it has a somewhat weird scale where a right angle is 90 degrees and a complete circle is 360 degrees. Whose idea was that, anyway? The Egyptians’, who took the Mesopotamians’ obsession with the number 60 and divided a circle into six 60-degree slices, no doubt to produce equal portions of the somewhat-dry Egyptian pizza.
60 is actually quite a cool number because it has so many factors; it can be evenly divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and, of course, 60. And the 360-degree circle is cool, too, because a right angle has the nice integer measure of 90 degrees, and the diagonal of a square is at a 45-degree angle (no decimals! Thank you…). And then there’s the 30-60-90 triangle, which has a certain charm, too. Look at all those clean, round numbers — a math student’s delight.
Enter the radian, which seems determined to be as difficult to work with as possible. All those nice round numbers? Gone. A complete circle is 6.28318531 radians. A right angle? 1.57079633 radians. Oh, and those decimals go on literally forever.
You probably know that these weird values stem from the fact that radians are built on the value of 
; the central angle of a complete circle is , a right angle is , and so on. It’s all clearly part of a global conspiracy to torment high school students and to ensure they don’t get too comfortable with trigonometry.
Degrees seem so much more civilized, but it’s true that the degree system is utterly arbitrary; there’s nothing special about the number 360 that makes it uniquely suited to being the measure of the central angle of a circle, though it has some nice qualities in that regard as mentioned earlier. It’s just a number that was chosen 6,000 years ago, and that’s that; there’s no magic mathematical relationship from which it stems.
Radians, however, represent the result of some brainiacs actually devising the perfect, natural measure for angles; though the general idea had been around for a few hundred years, the final version of the concept was hit upon by Roger Cotes in 1714, though the term “radian” wasn’t employed until more than a hundred and fifty years later, in 1869. And what’s so “natural” about the radian? It’s simple: one radian is the central angle of a circle that intercepts an arc that is one radius in length. That is, the number of radians in a central angle is the same as the number of radiuses of arc the angle intercepts.
This definition means we can completely eliminate the messy math associated with figuring out arc lengths when given central angles measured in degrees, and vice-versa. If we have a central angle of 2 radians, the corresponding arc length is 2 radiuses; if we have an arc with a length of radiuses, that means the central angle is radians. Compare that with what’s involved in making those same determinations when working in degrees; for a circle of radius and a central angle in degrees , you can use a proportion that relates arc ’s fraction of the complete circumference of to the angle ’s fraction of a complete circle’s central angle of 360 degrees:
Pretty unpleasant stuff! Here’s what that looks like:
So, for example, if the central angle is 100 degrees and the radius is 5, you’d have this:
But with radians, as explained a bit earlier, the arc length in radiuses is the same as the central angle in radians. Here’s the general case for a central angle and a radius :
If the central angle is 1 radian, the arc length is just 1 radius:
If the radius is 5, for example, the arc length will also be 5:
The power of the radian is also evident when the arc length is known, and the central angle needs to be determined; here, for example, the arc is equal to 2 radiuses, so the central angle is simply 2 radians (go ahead and compute the central angle in degrees for that situation — you’ll be breaking out the calculator):
These examples should make clear that radians are your friend, and you should consider asking them out to the movies, or at least hoping that they show up on your next SAT. They make your life easier once you understand and embrace them, so if you were a radian-hater before, we hope you give them another chance.
