You can set the height of your page designer layout to be 1/3 the height of a normal piece of paper. Then, when you go to print, choose to print on a normal size of paper, and choose the option to print multiple records. That will give you the effect that you’re looking for.
For even more advanced options when printing documents (including automating the printing of your documents), you can also turn to DocuMint. But it sounds like your needs are simple enough that you can just stick with Airtable’s Page Designer.