How was the area formula for a circle ($A = \pi r^2$) derived before the introduction of calculus? How did mathematicians prior to the coming of calculus derive the area of the circle from scratch, without the use of calculus?
The area, $A$, of a circle is $\pi r^2$. Given radius $r$, diameter $d$ and circumference $c$, by definition, $\pi := \frac cd$.

By trial and error and by numerical approximations. The ancient mathematicians (Babylonians, 1800 BC) tried to square a circle (approximating the area of circle with a square, constructing a square with the same area as a circle, proved impossible in 1882 AD), to calculate $\sqrt{2}$, $\pi$, etc.
The precise calculation of the area of circle requires an infinitesimal analysis (calculus).
