Calculate Trapezoid Area with Equilateral Triangle and Parallelogram Properties

To find the area of trapezoid $ABCD$, we need to determine the height using $\triangle EBC$, which is equilateral with side $BC = 3$ cm.

• Step 1: Calculating height of $\triangle EBC$.
  For $\triangle EBC$, the height ($h_t$) is $h_t = \frac{\sqrt{3}}{2} \times 3 = \frac{3\sqrt{3}}{2}$ cm.
• Step 2: Confirm equal base length.
  The base $AB$ is considered equal to $ED$ in parallelogram $ABED$, it is shared in the trapezoid.
• Step 3: Use trapezoid area formula $A = \frac{1}{2} \times (b_1 + b_2) \times h$.
  Considering $b_1 = AB = 3$ cm (since $BE = BC$) and $b_2 = 9$ cm (given $DC$), with the height $h$ same as equilateral triangle $EBC$, calculate:

The exact calculation becomes:

$A = \frac{1}{2} \times (3 + 9) \times \frac{3\sqrt{3}}{2} = \frac{1}{2} \times 12 \times \frac{3\sqrt{3}}{2} = 18\sqrt{3}$ square centimeters.

Approximating, $18\sqrt{3} \approx 27.3$ cm².

Therefore, the area of trapezoid $ABCD$ is $\boldsymbol{27.3}$ cm².