What is the distance of the Earth's orbit around the Moon? I realize the Earth and the Moon both orbit around their shared center of mass, and that in the case of the Earth and Moon this center is "inside" the Earth. However, I'm looking for an approach to determine the distance the Earth travels in one "orbit around the moon" (as small as it may be).
The mass of the Earth is about 81 times that of the Moon and the distance between their centres of mass is typically 384 400 km, so the centre of mass will be about 1/82 of this distance from the centre of mass of the Earth, which is about 4700km. 

The orbits are not too far from circular, so the Earth travels about $2 \pi$ times that each month, so a little under 30000km.