The surface area (SA) of a rectangular prism can be calculated using the formula:
\[ SA = 2(lw + lh + wh) \]
where \( l \) is length, \( w \) is width, and \( h \) is height. We are given that the height \( h \) and width \( w \) are both 9 cm, and the surface area is 432 cm². We need to find the missing length \( l \).
Substituting the known values into the surface area formula:
\[ 432 = 2(l \cdot 9 + l \cdot 9 + 9 \cdot 9) \]
Simplifying this:
\[ 432 = 2(9l + 9l + 81) \] \[ 432 = 2(18l + 81) \] \[ 432 = 36l + 162 \]
Now, subtract 162 from both sides:
\[ 432 - 162 = 36l \] \[ 270 = 36l \]
Next, divide both sides by 36 to solve for \( l \):
\[ l = \frac{270}{36} \]
Calculating this gives:
\[ l = 7.5 \]
Thus, the missing length of the rectangular prism is:
\[ \boxed{7.5} \text{ cm} \]