6. (10 points) As shown in the figure, $A F=7 \text{ cm}, D H=4 \text{ cm}, B G=5 \text{ cm}, A E=1 \text{ cm}$. If the area of quadrilateral $E F G H$ inside square $A B C D$ is $78 \text{ cm}^{2}$, then the side length of the square is ( ) $\text{cm}$.
A. 10
B. 11
C. 12
D. 13
【Analysis】The area of quadrilateral $E F G H$ = the area of square $A B C D$ - the areas of the four small triangles; Let the side length of square $A B C D$ be $x$, then the side lengths of the four small triangles are all determined;

Set up an equation to solve for $x$.
【Solution】Solution: $S_{\text {quadrilateral } E F G H}=S_{\square A B C D}-S_{\triangle A E F}-S_{\triangle F B G}-S_{\triangle C G H}-S_{\triangle D H E}=A B \times B C-A E \times A F$
$$
\begin{array}{l}
\div 2-B G \times B F \div 2-G C \times G H \div 2-D E \times D H \div 2=x^{2}-7 \times 1 \div 2-5 \times(x-7) \div 2-(x-5) \\
\times(x-4) \div 2-4 \times(x-1) \div 2=78 .
\end{array}
$$

Simplify $x^{2}=144$;
Therefore, the answer is: $C$.