The volume of the Moon is about 21.9 billion cubic kilometers.
Does that sound like a lot? For comparison, the volume of the Earth is 1 trillion cubic kilometers. In other words, the volume of the Moon is only 2% the volume of the Earth.
The interior of the Moon is composed of a crust, mantle and core. Astronomers think that the lunar core is about 350 km across, and accounts for only 20% of the size of the Moon. Most other planets and moons in the Solar System have a core that accounts for about 50% of their diameter. Outside the core is the middle mantle, and this is surrounded by an upper mantle.
You can listen to a very interesting podcast about the formation of the Moon from Astronomy Cast, Episode 17: Where Did the Moon Come From?