Answer
$(2-\sqrt{2})\pi^2$
Work Step by Step
For any matrix \[
\left[ {\begin{array}{cc}
a & b \\
c & d \\
\end{array} } \right]
\] the determinant equals $ad-bc$. For the matrix \[
\left[ {\begin{array}{cc}
\pi & \pi^2 \\
\sqrt{2} & 2\pi \\
\end{array} } \right]
\] the determinant equals $\pi(2\pi)-(\sqrt{2})(\pi^2)=(2-\sqrt{2})\pi^2$.