Q:

Why is the area of a circle $ \pi r^2$ and not $2\pi r^2$?

If you take the circumference of the circle and stretch it into a rope and then multiply it by the height(radius) why don't you get the area of a circle?
I ask this question when using the shell method revolved around the $x$ axis on $x^2$ from $[0,2] $ instead of the disk method.

A:

Because if you join the ends of a small piece of your rope to the centre, you get a triangle, not a rectangle. A triangle has half the area of a rectangle with the same base and height.

