Someone took a huge square map and cut it into identical rectangular pieces by cutting straight down even columns and across even rows, thus preserving the orientation (portrait or landscape mode) of each piece identically, and then bound the pieces together to make an atlas, with each piece becoming a page. Each piece was an integral number of inches both vertically and horizontally.

There were between 50 and 150 pieces altogether, and it turned out that on each page, over 50% of the area was within two inches of the edge of the paper. If either dimension (height or width) of the page had been just one inch larger it would have been no longer true that over 50% of the page was within two inches of the edge.

What was the size of each piece that became a page?