To find the missing length (let's call it \( l \)) of the rectangular prism given the height (\( h \)) and width (\( w \)) are both 9 cm and the surface area (\( SA \)) is 432 cm², we can use the formula for the surface area of a rectangular prism:
\[ SA = 2(lw + lh + wh) \]
Substituting the values we know:
- \( w = 9 \) cm
- \( h = 9 \) cm
- \( SA = 432 \) cm²
We can substitute these into the surface area formula:
\[ 432 = 2(9l + 9l + 9 \cdot 9) \]
This simplifies to:
\[ 432 = 2(18l + 81) \]
Now divide both sides by 2:
\[ 216 = 18l + 81 \]
Next, we subtract 81 from both sides:
\[ 216 - 81 = 18l \]
\[ 135 = 18l \]
Now, divide by 18 to find \( l \):
\[ l = \frac{135}{18} = 7.5 \]
Thus, the missing length of the rectangular prism is:
\[ \boxed{7.5 \text{ cm}} \]