To find the missing length of a rectangular prism given the height and width, we can use the formula for the surface area of a rectangular prism:
\[ \text{Surface Area} = 2(lw + lh + wh) \]
where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height.
Given that the height (\( h \)) and width (\( w \)) are both 9 cm, we can substitute these values into the formula:
\[ \text{Surface Area} = 2(l \cdot 9 + l \cdot 9 + 9 \cdot 9) = 2(9l + 9l + 81) \]
This simplifies to:
\[ \text{Surface Area} = 2(18l + 81) = 36l + 162 \]
We know the surface area is 432 cm², so we can set up the equation:
\[ 36l + 162 = 432 \]
Now, we'll solve for \( l \):
- Subtract 162 from both sides:
\[ 36l = 432 - 162 \] \[ 36l = 270 \]
- Divide both sides by 36:
\[ l = \frac{270}{36} \] \[ l = 7.5 \]
Thus, the missing length of the rectangular prism is:
\[ \boxed{7.5} \text{ cm} \]