Given:
- ABCD is a rectangle.
- Points E and F are on side AB such that they trisect AB.
- The difference between the areas of trapezium CDEF and the triangle ADE is 12 sq. cm.

To find:
- The area of rectangle ABCD.

Solution:

Step 1: Understanding the problem

Let's start by visualizing the problem. We have a rectangle ABCD, and points E and F are trisecting side AB. This means that AE = EF = FB. We need to find the area of rectangle ABCD.

Step 2: Breaking down the problem

To solve this problem, we can break it down into two parts:
1. Finding the area of trapezium CDEF.
2. Finding the area of triangle ADE.

Step 3: Finding the area of trapezium CDEF

Since trapezium CDEF is formed by the rectangle ABCD, we can find its area by subtracting the area of triangle ADE from the area of the rectangle ABCD.

Let the length of AB be 'a' and the width of the rectangle be 'b'.
The length of AE, EF, and FB is a/3.

The area of trapezium CDEF can be calculated as:
Area of trapezium CDEF = Area of rectangle ABCD - Area of triangle ADE

Step 4: Finding the area of triangle ADE

The area of a triangle can be calculated using the formula:
Area of triangle = (1/2) * base * height

In triangle ADE, the base is a/3 and the height is b.

Step 5: Calculating the area of trapezium CDEF

Using the formula for the area of a rectangle, we can calculate the area of rectangle ABCD:
Area of rectangle ABCD = length * width = a * b

Using the formula for the area of a triangle, we can calculate the area of triangle ADE:
Area of triangle ADE = (1/2) * base * height = (1/2) * (a/3) * b

Finally, we can calculate the area of trapezium CDEF by subtracting the area of triangle ADE from the area of rectangle ABCD:
Area of trapezium CDEF = Area of rectangle ABCD - Area of triangle ADE
Area of trapezium CDEF = a * b - (1/2) * (a/3) * b

Step 6: Solving the equation

According to the given information, the difference between the areas of trapezium CDEF and triangle ADE is 12 sq. cm:
Area of trapezium CDEF - Area of triangle ADE = 12

Substituting the expressions for the areas of trapezium CDEF and triangle ADE, we get:
(a * b - (1/2) * (a/3) * b) - (1/2) * (a/3) * b = 12

Simplifying the equation, we get:
(a * b - a * b/6) - a * b/6 = 12

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