What Is A Radian?

In geometry, a radian is a unit used to measure angles. The radian comes from the length of the circle's radius. The segment of a circle's circumference that corresponds to the angle made by two radius lines makes an arc. The angle that this arc creates, when you draw lines from its starting and end points to the circle's center, is one radian. Though the radian might appear odd and complicated at first, it simplifies equations in math and physics.

Outside of physics and advanced math, degrees are typically more familiar units for angular measurements. A circle, for example, has 360 degrees, a triangle has 180 and a right angle has 90. By contrast, a full circle has 2 × π (pi) radians, a triangle has π radians and a right angle is π/2 radians. A circle has a whole number of degrees, whereas in radians the value is an irrational number, so radians at first blush might seem strange. On the other hand, you might express fractions of a degree as a decimal, or as the minutes, seconds and decimal seconds you also use with time, so the degree has issues of its own.

Degree measurements are typically easier to deal with than radians for basic arithmetic and trigonometry; you seldom have to deal with fractions of π when expressing an angle. But for calculus and other advanced math, it turns out that radians are easier. For example, the power series for the sine function in radians is as follows:

\(\sin(x) = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \frac{x^9}{9!} + ...\)

In degrees, the function looks like this:

\(\sin(x) = (π × x/180) – \frac{(π × x/180)^3}{3!} + \frac{(π × x/180)^5}{ 5!} – \frac{ (π × x/180)^7}{7!} + \frac{(π × x/180)^9}{9!} + ...\)

For this power series, note that you need to repeat the "π × ​x​/180" for every term – a lot of extra writing and calculation compared to the neater, more compact equivalent in radians. The radian comes from the natural geometry of a circle rather than a division by an arbitrary number, as degrees do. Because radians make many calculations easier, mathematicians think of the unit as more "natural" than degrees.

In addition to power series such as the sine-function example, you'll see radians in math involving calculus and differential equations. For example, when you use radians, the derivative of the sine function, sin(​x​), is simply the cosine, cos(​x​). In degrees, however, the derivative of sin(​x​) is the more cumbersome (π ÷ 180) × cos(​x​). As you progress in math, the problems get harder, and the solutions require many more lines of calculation and algebra. Radians save you a lot of unnecessary extra writing and reduce the chances of making mistakes.

In physics, formulas for the frequency of waves and the rotational speed of objects use a lower-case omega, "​ω​," as a convenient shorthand for "2 × π × radians per second."

The formulas to convert degrees to radians and back again are straightforward. To convert angles in degrees to radians, multiply the angle by π, and then divide by 180. For example, a circle has 360 degrees. Multiplied by π, that becomes 360π; then divide by 180, and you get 2π radians. To convert from radians to degrees, multiply by 180, and then divide by π. For example, convert a right angle, π ÷ 2 radians. Multiply by 180 to get 90π, and then divide by π to get the result, 90 degrees.