The given radius of the circle is \(3.1 \times {10^4}\;{\rm{cm}}\).

The area of a circle represents the region covered by a circular object in the 2-D plane. In the standard international system, the area of a circle is measured in meter square.

The relation for the area of a circle is \(A = \pi {r^2}\).

Substituting the variables by their values in the above relation, you get

\(\begin{aligned}{l}A = \pi {\left( {3.1 \times {{10}^4}\;{\rm{cm}}} \right)^2}\\ = 3.01 \times {10^9}\;{\rm{c}}{{\rm{m}}^2}.\end{aligned}\)

Uncertainty represents the range within which the actual value of measurement lies.

The relation for uncertainty in the area is

\(\frac{{\Delta A}}{A} = \frac{{2\left( {\Delta r} \right)}}{r}.\)

Here, \(\Delta r\) is the smallest possible uncertainty of the radius with the value \(0.1 \times {10^4}\;{\rm{cm}}\).

Substitute the variables by their values in the above relation, you get

\(\begin{aligned}{l}\frac{{\Delta A}}{A} = \left( {\frac{{2\left( {0.1 \times {{10}^4}\;{\rm{cm}}} \right)}}{{\left( {3.1 \times {{10}^4}\;{\rm{cm}}} \right)}}} \right)\\ \Rightarrow \frac{{\Delta A}}{A} = 0.0645 \times 100\% \\ \Rightarrow \frac{{\Delta A}}{A} = 6.45\% .\end{aligned}\)

Thus, the area of the circle and the approximation uncertainly are \(A = 3.01 \times {10^9}\;{\rm{c}}{{\rm{m}}^2}\) and \(\frac{{\Delta A}}{A} = 6.45\% \), respectively.