The definition of a radian is a unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius. If someone had never heard of a radian I would explain in to them as the angle created by the length of the radius around the circle. The unit circle is related to radians because you can measure there distance and find the lengths of the triangle in radians or degrees. The formula for the circumference of a circle and radian measure relate because they both deal with π and radius. Radians and degrees are equal, you can covert them to each other by multiplying by 180/π or π/180. I prefer degrees because they are easier to deal with and I am more familiar with them. However, radians seem more mathematically "pure" because of a circles close relation with π.
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