Answer
No, It is not possible to determine whether the area of the circle is $\pi r ^{2}$ or 2 $\pi r ^{2}$
Work Step by Step
We know that the dimension of both $\pi r^2$ and $2\pi r^2$ is $L^2$. The factor $2$ is a dimensionless constant and therefore dimensional analysis cannot determine whether the area of a circle is $\pi r^2$ or $2\pi r^2$.
