Integral Calculus, Circumference of a Circle

Circumference of a Circle

Review the definition of sine and cosine.  Recall that an imaginary particle moves around the unit circle counterclockwise, at constant speed, and the x and y coordinates define the cosine and sine functions respectively.  Since the speed is always 1, the length traveled is identical to the travel time.  A circuit is completed when t = 2π, hence the circumference is 2π.  (See arc length for more details.)  Scaling all variables, the circumference around a circle of radius r is 2πr.