Originally Posted by
HallsofIvy Here is one way to do it. Imagine dividing the circle into n "wedges". Now, line the wedges up so that the first has its "tip" upward, the second downward, etc., reversing the direction on each one. We get approximately a rectangle with the arcs of the circle forming both upper and lower edges, with the approximation becoming better for larger n. That "rectangle" has the radius of the circle as height and half the circumference as width (since the circumference formed both top and bottom). Since the circumference is $\displaystyle \pi d= 2\pi r$, half the circumference is $\displaystyle \pi r$. The area of the "rectangle", and so the area of the circle, is "height times width" $\displaystyle = r(\pi r)= \pi r^2$. In the limit, as n goes to infinity, that "approximation" becomes exact. Since the true area of the circle does not depend on n, that formula gives the true area.
Notice that Captain Black is defining "$\displaystyle \pi$" to be the area of a circle of radius 1 while I am taking the definition of "$\displaystyle \pi$" as the ratio of the circumference of a circle to its diameter.