At first, let us look for the forces acting on the moon. The Moon rotates around the Earth, so centrifugal force is exerted radially outward of the orbit. Gravitational force is also exerted radially inward of the orbit. That is, the centrifugal force and gravitational force point opposite to each other. Since there is no radial acceleration, these forces must be equal in magnitude.Then, we get the following equations:
We can calculate the speed of the Moon.
The speed is also the same as the orbit radius times angular velocity.
This answer is a rough estimation of the radius. The actual mean distance to the Moon from the Earth is 384,400km.