To find the missing length (let's call it \( l \)) of a rectangular prism with height (\( h \)) and width (\( w \)), the formula for the surface area (\( SA \)) of a rectangular prism is:
\[ SA = 2(lw + lh + wh) \]
Given:
- Height \( h = 9 , \text{cm} \)
- Width \( w = 9 , \text{cm} \)
- Surface area \( SA = 432 , \text{cm}^2 \)
Plugging the known values into the surface area formula:
\[ 432 = 2(9 \cdot 9 + 9 \cdot l + 9 \cdot l) \]
This simplifies to:
\[ 432 = 2(81 + 9l + 9l) \] \[ 432 = 2(81 + 18l) \] \[ 432 = 162 + 36l \]
Subtract 162 from both sides:
\[ 432 - 162 = 36l \] \[ 270 = 36l \]
Now, divide both sides by 36 to find \( l \):
\[ l = \frac{270}{36} = 7.5 \]
Thus, the missing length of the rectangular prism is:
\[ \boxed{7.5} , \text{cm} \]