What type of glue is used for gluing acrylic pieces together?
I would like to expand on EmRoBeau's answer a little. Acrylic cement is the common product used when trying to merge pieces of acrylic together. It is important to note that while it is commonly referred to as a glue it does not function as one. It is actually a solvent that fuses or welds the plastics components together. 

It is really important that your pieces lie together and flat with no seems or gaps. This could also be called perfectly mated. Acrylic cement only makes a strong bond when there is no space between the merged acrylic and the actual fluid. Since you are not joining on edges since, in theory, should be of less concern as the sheets should be manufactured flat. 

I would caution that you try this on progressively larger pieces though. If an incorrect amount of fluid is used it will make for a poor weld and the project would be potentially ruined.

A good primer video would be one from Tap Plastics. While it does promote a product there is more useful information beyond that.


Introduction to acrylic cement

Since the product you are working with is a chemical solvent use the proper protection measures as described on the labeling.