To find the missing length of the rectangular prism, we start by recalling the formula for the surface area \( SA \) of a rectangular prism, which is given by:
\[ SA = 2(lw + lh + wh) \]
Where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.
In this case, we know:
- Height \( h = 9 \) cm,
- Width \( w = 9 \) cm,
- Surface area \( SA = 432 \) cm\(^2\).
We need to find the length \( l \). Substituting the known values into the surface area formula, we have:
\[ 432 = 2(9 \cdot 9 + 9 \cdot l + 9 \cdot l) \]
Now simplify the expression inside the parentheses:
\[ 432 = 2(81 + 9l + 9l) \] \[ 432 = 2(81 + 18l) \]
Distributing the 2:
\[ 432 = 162 + 36l \]
Next, we isolate \( l \) by first subtracting 162 from both sides:
\[ 432 - 162 = 36l \] \[ 270 = 36l \]
Now, divide both sides by 36 to solve for \( l \):
\[ l = \frac{270}{36} \] \[ l = 7.5 \]
Thus, the missing length of the rectangular prism is
\[ \boxed{7.5} \text{ cm}. \]