Teaser 1973: Straw store

From The Sunday Times, 9th July 2000 [link]

Farmer Fermat has more straw than he can safely stack on his usual rectangular storage area. So he extends the shorter sides of the area to equal the longer sides, thereby forming a square. Alongside he adds another square area, whose side is equal to the shorter side of the original rectangle. The total area of the two squares together is 243 square feet larger than the original rectangle.

He can now stack straw on each square, up to a height equal to the length of the side of each square, effectively forming two cubes, and the total volume in cubic feet is a perfect cube.

What was the perimeter of the original rectangle?