A parallelogram has sides A, B, C, and D. Sides A and B have a length of #6 # and sides C and D have a length of # 4 #. If the angle between sides A and C is #(7 pi)/12 #, what is the area of the parallelogram?

The area of the parallelogram can be calculated using the formula: Area = base × height. The base of the parallelogram can be taken as side A or side C, and the corresponding height would be the perpendicular distance between these sides. Since sides A and C are adjacent and form an angle, the height can be determined using trigonometry. Specifically, the height can be found using the formula: height = AB * sin(θ), where AB is the length of side B (or side D) and θ is the angle between sides A and C. Once the height is determined, the area can be calculated by multiplying the base by the height.

To find the area of the parallelogram, we can use the formula:

Area = base * height

In a parallelogram, the height is the perpendicular distance between two parallel sides. Given the lengths of sides A, B, C, and D, and the angle between sides A and C, we can find the height using trigonometric principles.

First, let's find the height using the formula for the height of a parallelogram:

Height = side * sin(angle)

We'll use side A as the base and side C as the corresponding height. So, plugging in the values:

Height = 6 * sin((7π)/12)

Now, we need to find the area:

Area = base * height

Area = 6 * (6 * sin((7π)/12))

Area ≈ 6 * (6 * 0.9659)

Area ≈ 6 * 5.794

Area ≈ 34.764 square units

So, the area of the parallelogram is approximately 34.764 square units.