1. **State the problem:** We are given an area expression $81x^2$ cm$^2$ and a length of 3 cm, and we want to find the missing length of the rectangle. 2. **Formula used:** The area $A$ of a rectangle is given by the formula: $$A = \text{length} \times \text{width}$$ 3. **Apply the formula:** Here, the area is $81x^2$ cm$^2$ and one side (length) is 3 cm. Let the unknown side be $w$ cm. $$81x^2 = 3 \times w$$ 4. **Solve for $w$:** $$w = \frac{81x^2}{3}$$ 5. **Simplify the fraction:** $$w = \frac{\cancel{81}x^2}{\cancel{3}} = 27x^2$$ 6. **Final answer:** The missing length is $27x^2$ cm. This means the rectangle's dimensions are 3 cm by $27x^2$ cm, giving the area $81x^2$ cm$^2$ as expected.