All of that is handled in the Logical Volume Manager. It's been a while since I worked with it, but I think in this particular setup, it's going to be a bit of a pain to extend. If you run pvs I would bet that /dev/sda3 is listed as the physical volume, which means it only sees that 60GB of space. I can't recall what the solution was here, but I remember that dealing with it was rather irksome.
You might be able to create /dev/sda4 with all of the remaining space on the drive, add /dev/sda4 as a physical volume, and then to the volume group. Once it's part of the volume group, you can extend the logical volume, and resize the file system.
You will want to read up on the Logical Volume Manager commands. It can be a little weird to wrap you head around at first, but once you get the hang of it, it's not so bad.
