THE MATH WAY
1. Complete the circle that the rounded corner would make if it kept turning.
2. Calculate the area of that circle like so:
a. Draw a straight line connecting two points of the circle, I think it's called a cord. It doesn't matter if it's directly in the center or not. Call this line A.
b. Mark the midpoint of A.
c. Draw a perpendicular line through the midpoint of A, connecting what will now be two opposite ends of the circle. Call this line B. I eyeballed this, so it's a little off ;/
d. Line B is now your diameter. Half of it is the radius, the midpoint is in the absolute center of the circle. Area is pi * r * r.
3. Draw a square that barely touches (is tangent to) the circle on all of the square's sides.
4. Area of the square is side * side, subtract the area of the circle to get the area between the circle and square.
5. Divide #4's area by 4. Exactly 1/4 of the area from number 4 is outside the curve, yet within the square.
6. Multiply #5 by 4 for the 4 rounded corners, and subtract that from the area of the shape as if it was a rectangle with points instead of curves. Ummm, I guess you could leave this and #5 alone since they basically cancel each other out, but it's good to show my work...I guess...
Rankeri is almost right about the area of an ellipse: Pi * A * B. But A is
half the length and B is
half the height, just like the radius is
half the diameter.
Yea...I'm kinda bored right now...