bad math, sod and Nightcrawler (not worms)

Because of my bad math I ordered 225 square feet of sod instead of the 300 square feet I needed. I realize this mistake derives from my errant belief that there's a constant ratio between perimeter and area. My reasoning went like this. If I measure the perimeter of the yard (it's an irregular shape) I should be able to rearrange the perimeter into a different configuration and then get the correct square footage. For example, if the perimeter of the yard is 10 feet, then I can rearrange those 10 feet into a rectangle and figure out the square footage, kind of like a poor man's calculus. There's one problem  with this: there is no constant relationship between perimeter and area. A 10 foot perimeter could be arranged into a 4 x 1 grid (8 feet on the sides, 2 feet on the ends) and equal 4 square feet. Or it could be arranged into a 3 x 2 grid (6 feet on the sides, 4 feet on the ends) and equal 6 square feet. After doing some research I discovered that if the circumference of a circle is 10 feet, the area is 7.9 square feet. Three ways of arranging a perimeter of 10 feet gives you three different answers: 4, 6, and almost 8 square feet.

Interestingly, my math error was compounded by another math error which prevented the mistake from being even larger. I think what I did was the equivalent of halving the perimeter. For example, if the perimeter of the yard was 10 feet, I looked for the nearest easy multiplier to make 10 which in this case would be 5 x 2. So I multiplied the two together. In the example, this makes 10 square feet. Whatever I did for the yard, it ended up about 75 square feet short.

A simple thought experiment would have proven the error in my reasoning. Imagine a rectangle that is about 5 feet wide and only a hair thickness tall. Obviously, this isn't an accurate measure of the area of a 10 foot perimeter.

Still, I find the lack of a correspondence between perimeter and area baffling. I took a string and arranged it in different ways. I tried making rectangles and circles and I felt like primitive man as I watched the area change dramatically from one configuration to another. At some level, I still don't understand how this works.

There's some part of my mind that incorrectly assumes certain inviolable constants in math and physics, like, apparently, the relationship between perimeter and area. This also explains the X-Men physics problem that bewilders me but not people who've studied physics. The problem goes like this. Nightcrawler can teleport from any one place to another. So, to save himself from falling, he might teleport himself closer to the ground. Apparently, people who know physics find this makes total sense. To me, however, it seems like gravity would be a constant lurking in the background. Nightcrawler could teleport closer to earth, but in the act of teleportation, gravity would be conserved: Nightcrawler would only teleport more quickly to his doom.

Below: both shapes have 10 foot perimeters but different areas.