Packing

I have a 21" square tray that holds seven glasses in a row, along any one edge, as illustrated. (The glasses are exactly three inches in diameter, perfectly cylindrical, and very tall.) How many glasses can the tray hold?

The correct answer is NOT 49.

You would think a tray twice as wide and twice as long would hold four times as many glasses. Not so.

If a square tray holds 14 glasses in a row along any one edge, how many will it hold?

Hint: The exact size of the glasses is unimportant. Try this with coins, to simulate the bottoms of the glasses. How many coins can you pack inside a square that is seven or fourteen times the diameter of one coin?