"so much of the perimeter curled up at the edges", is it? If it's true of my circle, it's also true of your square. Those four "straight" lines you used to box in the circle? They are all kinds of jagged and crooked too, making each an unknown distance greater than the theoretical unit length. They might not even be equal lengths, for all we know. In fact, they could be infinitely long. We cannot know the true length of anything, including your collapsing corners box, and thus all geometry and trigonometry is useless.

Except...we know that in the real, physical world, geometry and trig are the opposite of useless. We know through experiments and observation that the objects we agree to call "circles" have a perimeter that measures 3.14159...... times the measured diameter of those objects. We use that number to calculate how much sheet metal it will take, when rolled into a cylinder, to create a drum of a desired diameter, and viola! The drum thus formed does indeed have the desired diameter! We use the same 3.14159..... times radius times radius to calculate how much area is enclosed by these objects we agree to call "circles" and when we check with (for example) liquid in a drum, guess what? There is as much liquid in the drum as the math said there would be! Whereas, if you take 4 as your value of Pi nd calculate the amount of sheet metal to use for a given diameter drum, when you build it your drum will have a diameter larger than you wanted.

**BECAUSE PI ISN'T 4!!**

Then when you do your volume calculation with the actual diameter and Pi=4, you will find your drum cannot hold the amount you calculated.

**BECAUSE PI STILL ISN'T 4!!!**

Didn't you do these very tests when you were a child in school? In my class we each were given a different length piece of construction paper. We measured its length, formed it into a circle by taping the ends, then measured its diameter. The whole class then reported their numbers, which the teacher wrote on the board. She then calculated the ratios, to demonstrate that every circle had the same ratio (give or take the measuring skill of children of course). We then filled the cylinders with a single layer of peas, and counted them as a rough measure of area. Again the numbers were called out to teacher, who applied Pi R squared to prove the rule, again subject to the imperfection of school children's construction.