 So up until now, you've probably been measuring angles in degrees. And that's like using training wheels or drinking out of a sippy cup or measuring things in feet and inches. Sooner or later, you have to take off the training wheels, drink from regular cups, and use the metric system. And so let's recap how we measure an angle. When measuring an angle, the natural measure is the turn. But it's too big. The deplorable measure is the degree, but it's meaningless. The best and most useful measure is the radian, and that comes about as follows. Given an angle, we can make the vertex the center of a circle, and for convenience, we'll make the circle have a radius of one, and this allows us to measure the angle in another way. The radian measure of an angle is the distance between the legs as measured a longer unit circle centered at the vertex of the angle. For example, an angle has a measure of a quarter turn. What's its radian measure measured counterclockwise? Since we don't want to make this problem harder than it has to be, we'll draw a picture. For the radian measure, we'll draw a unit circle centered at the vertex. We'll draw an angle that corresponds to a quarter turn rotation. Since the angle has a measure of a quarter turn, we'll walk one quarter of the way around the circle. Now, we need to know how far we've walked, so we want to find the circumference of a circle with a radius of one. So the circumference of any circle is 2 pi times the radius, and since the radius is one, our circumference will be We've walked one quarter of the way around the circle, so we've walked one quarter of the circumference, and so that distance is one half pi. Now, something worth keeping in mind is that you can say the angle measure is one half pi. If you're using radians, you don't need to specify radians, but you can if you want to. Or we might convert into radian measure, two full turns, 30 degrees. Since one turn is once around the circle, and the distance we'd have to go to go once around the circle is 2 pi, then one turn is 2 pi radians, so two turns will be 4 pi radians. Since 360 degrees is once around the circle, we know that 360 degrees must be 2 pi radians, and for variety's sake, we won't specify radians this time because we don't have to. Now, if I want to get 30 degrees, let's find one degree. Let's divide through by 360, and so we know that one degree is 2 pi over 360 radians. Well, that was useless. I didn't want one degree. I wanted 30 degrees. Oh wait, I can get 30 degrees by multiplying by 30, and I get 30 degrees is equal to this product, which we can simplify to 1 6 pi radians.