In mathematics outside of school we use radians instead of degrees for convenience. I have found that many undergraduate students don’t understand what a radian is.
An angle of 1 radian is created when the length of the arc created by the angle is equal to the length of the radius. Thus, a radian is the angle that occurs when the arc and radius are equal in length. This can be seen in the image below, where we have an angle of 1 radian, and the arc and radius are of the same length.
Speaking more mathematically, if we have a radius length denoted \(r\), then an angle of 1 radian occurs when that angle creates an arc length equal to \(r\). Expanding upon this notion, we discover that an angle of 2 radians gives an arc length that is twice the size of the radius. By increasing the angle, we get a larger arc length and subsequently a larger radian value because the arc is getting larger in comparison to the radius which is always fixed. On the other hand, by decreasing the angle, we get a smaller arc length and subsequently a smaller radian value because the arc is getting smaller in comparison to the fixed radius. From this we can say that the arc length, \(\alpha\), is equal to the angle in radians, \(\theta\), multiplied by the radius, \(r\).
But why is a 360 degree angle equivalent to \(2\pi\) radians? Note that the circumference of a circle is equal to \(2\pi r\). This means that the length of the arc of a full circle is \(2\pi r\). That is “two multiplied by the value of pi multiplied by the value of the radius”. But since we know that the arc length is equal to the angle in radians multiplied by the radius, we can do some algebra to find what the angle will be when the arc length is equal to \(2\pi r\).
As can be seen from above, we have proven that an angle of 360 degrees is equivalent to \(2\pi\) radians and also gained some understanding and intuition about what radians really are.