How would I derive Kepler's third law from Newton's law of gravitation and the equations of circular motion?

Kepler's third law states that the square of the period of the orbit is directly proportional to the cube of the radius of the orbit (T^2=kr^3) where r is some constant to be determined. This can be determined using:

Newton's law of gravitation: F=GMm/r^2 Centripetal Force: F=mw^2r, where w is the angular velocity in rad/s.

By equating these 2 equations and cancelling out any terms possible we arrive at GM=r^3w^2. The angular velocity can be described as the angle a body has travelled through in a period of time. Assuming a full circular orbit this would be equal to 2pi radians in a period of T. Therefore w=2pi/T. This can be substituted in to obtain T^2=(4*pi^2/GM)*r^3. Therefore the constant of proportionality equals 4pi^2/GM.