To find the missing length of the rectangular prism, we can use the formula for the surface area of a rectangular prism:
\[ \text{Surface Area} = 2lw + 2lh + 2wh \]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height. Given that both the width \( w \) and the height \( h \) are 9 cm, we can plug these values into the formula. Let \( l \) be the missing length we are trying to find.
\[ 432 = 2(l \cdot 9) + 2(l \cdot 9) + 2(9 \cdot 9) \]
This simplifies to:
\[ 432 = 2(9l) + 2(9l) + 2(81) \]
\[ 432 = 18l + 18l + 162 \]
\[ 432 = 36l + 162 \]
Next, we subtract 162 from both sides:
\[ 432 - 162 = 36l \]
\[ 270 = 36l \]
Now, divide both sides by 36 to solve for \( l \):
\[ l = \frac{270}{36} \]
Calculating this gives:
\[ l = 7.5 \]
Thus, the missing length is
\[ \boxed{7.5} \text{ cm} \]