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, $\pi := \frac cd$.

Another way to derive the formula for the area of a circle is by decomposing it into several concentric rings (annuli) to form a "triangle" as seen in the below diagram.  As the width of the concentric rings approach zero, it forms a triangle of base circumference ($C$ = $\pi$$d$) and height $r$.

Since we know the area of a triangle is $\frac{bh}{2}$, by substitution $$\frac{\pi d\cdot r}{2}$$
Since $d$ = 2$r$, $$A = \frac{\pi \cdot 2r\cdot r}{2} \rightarrow A = \frac{\pi \cdot 2r^2}{2} \rightarrow A = \pi \cdot r^2$$.
