Answer
approximately $511,185,933$ $km^2$
Work Step by Step
Using the given formula, $
r=\sqrt{\dfrac{A}{4\pi}}
,$ with $
r=6378$ $km$, then
\begin{array}{l}\require{cancel}
6378=\sqrt{\dfrac{A}{4\pi}}
\\\\
(6378)^2=\left(\sqrt{\dfrac{A}{4\pi}}\right)^2
\Rightarrow\text{ (square both sides)}
\\\\
40678884=\dfrac{A}{4\pi}
\\\\
4\pi\cdot40678884=\dfrac{A}{4\pi}\cdot4\pi
\\\\
162715536\pi=A
\\\\
A=162,715,536\pi
\\\\
A\approx511,185,933
.\end{array}
Hence, the surface area, $A,$ of the earth is approximately $511,185,933$ $km^2$.