Here we have a circle and three sliders which should all be at 0. We're going to see where the definition of a radian comes from and how many radians fit into a circle.[br][br]First, grab the Radius slider and move all the way to the right. It should say Radius=5. You can see the measure of the [i]radius[/i] on the diagram by the little number under the line segment.[br][br]Next, grab the Angle slider and move it all the way to the right. It should say Angle=1. You can see that this angle is about 57.3[math]^\circ[/math]. Note that you can also see the arc length being swept out by the angle by the little number next to the arc. When the angle is wide enough that the [i]arc length[/i] is equal to the [i]radius[/i], that's an angle of 1[b] radian[/b].[br][br]Now, leave the Angle at 1 and grab the Radius slider. Move it back and forth. Notice that the [i]arc length[/i] and the [i]radius[/i] stay the same. A [b]radian[/b] is a radian is a radian, no matter how big or small a circle you're looking at.[br][br]Finally, move the Radius slider back to 5 and then grab the Rotate slider at the bottom. As you slide this all the way to the right, you'll see six sectors (pie slices) with a central angle of 1 [b]radian[/b] that don't quite make it all the way around the circle.[br][br]How [b]radians[/b] does it take to sweep all the way around a circle? We know that the [i]arc length[/i] of each of these sectors is equal to the [i]radius[/i] of the circle. We also know that it takes six-and-some-change of these [i]arc lengths[/i] to make a full circumference of the circle. How many [i]radii[/i] does it take to make a full circumference of a circle?[br][br][math]circumference=2\pi\cdot radius[/math][br][br]It take [math]2\pi[/math] [i]radii[/i] to make the circumference of a circle. In the same way, it take [math]2\pi[/math] [b]radians [/b]to sweep out a full circle. That means [math]2\pi=360^{\circ}[/math]. (Unlike degrees, angles measured in radians are typically given as just numbers without a unit symbol.)