Example 6 Given $a>0, b>0$. Find the area of the triangle with side lengths $\sqrt{a^{2}+b^{2}}$, $\sqrt{a^{2}+4 b^{2}}$, and $\sqrt{4 a^{2}+b^{2}}$.
(The 12th "Hope Cup" National Mathematics Invitational (Junior $\Rightarrow)$
Explanation: Directly using the formula for the area of a triangle is quite complicated, but by constructing the rectangular figure as shown in Figure 2, the area of the desired triangle can be easily obtained through the relationship between the whole and the parts. Let the area of the desired triangle be \( S \), then
\[
\begin{aligned}
S & =2 a \times 2 b-\frac{1}{2} a b-\frac{1}{2} a \times 2 b-\frac{1}{2} \times 2 a \times b \\
& =\frac{3}{2} a b .
\end{aligned}
\]