5. (10 points) As shown in Figure 1, the width of the “$L$” shape is 3 cm. Place 4 such “$L$” shapes in the four corners of a 3x3 grid, forming the shape as shown in Figure 2. If the total area of the 4 “$L$” shapes is exactly equal to the area of the shaded part in Figure 2, then the area of 1 “$L$” shape is $\qquad$ square centimeters.
【Solution】Solution: Let the side length of the smallest square in the figure be $a$, then the side length of the large square formed by 9 small squares is $3a$, and the side length of the slightly larger square than the small square is $a+3$,
The sum of the areas of 4 $L$-shaped figures is $4(a+3)^{2}-4a^{2}$, the area of the shaded part $=5a^{2}$,
According to the problem, $4(a+3)^{2}-4a^{2}=5a^{2}$,
$$
\begin{array}{l}
\therefore 4(a+3)^{2}=9a^{2}, \\
\therefore 2(a+3)=3a, \\
\therefore a=6,
\end{array}
$$
$\therefore$ The area of 1 "L" shape is $(6+3)^{2}-6^{2}=45$.
Therefore, the answer is: 45.