Calculate Trapezoid Height: Finding AE Given Area 18 cm²

To determine the length of side $AE$, we'll use the formula for the area of a trapezoid. The formula is:

$\text{Area} = \left( \frac{b_1 + b_2}{2} \right) \times h$

Given values are $b_1 = 3$ cm, $b_2 = 6$ cm, and $\text{Area} = 18$ cm$^2$.

Substituting these into the equation:

$18 = \left( \frac{3 + 6}{2} \right) \times h$

Simplify the expression:

$18 = \left( \frac{9}{2} \right) \times h$

Multiply both sides of the equation by 2 to eliminate the fraction:

$36 = 9h$

Now, solve for $h$ by dividing both sides by 9:

$h = \frac{36}{9} = 4$

Thus, the length of side $AE$ is 4 cm.

Therefore, the solution to the problem is $\mathbf{AE = 4 \, \text{cm}}$.