Return to trigonometry page

Radian Measure

Consider a circle with radius 1. What is its circumference? If you remember that the formula for the circumference is given by C=2πr, you should be able to quickly come up with the answer that the circumference is 2π.

Now suppose that we have only a semi-circle with radius 1. What is the measure of this arc? Since it's half of the circle, we can take half the circumference of the circle, and the measure will be π.

Now, how many degrees are there in a circle?

You should have come up with 360°

So how long is a one degree arc with radius 1?

For this, you should take the circumference of the circle with radius 1, 2π, and divide it by 360° That gives π/180.

Why do we care about this? Because by talking about an angle in terms of the arc length with radius 1, we can speak about angle measurements with a level of definiteness that's not possible with degrees: The choice of 360° to form a circle is one that is largely arbitrary (although of some convenience since we'll do a lot with angle measures of 0°, 30°, 45°, 60° and 90°. There's no reason that the circle couldn't be divided into some other units (in fact, there was an attempt at one point to divide circles into 400 units so that each quadrant would be divided into 100 parts). But we will see later in the class that talking about angles in terms of the measure of an arc with radius 1 has some added benefits in calculating the values of trig functions.

So if we talk about the measure of the arc with radius 1, we call these units radians. We've seen above that one degree is π/180 radians, so we can take any measure in degrees and multiply by π/180 to get the measure in radians. Similarly to convert from radians to degrees, we can divide by π/180. So, for example, if we have π/4 radians, we can divide by π/180 and find that this is equivalent to 45°.

Last updated 26 Feb 2006.   Send me an e-mail.