All of the lengths with one mark have length 5, and all of the side lengths with two marks have length 4. With this knowledge, we can add side lengths together to find that one diagonal is the hypotenuse to this right triangle:

Using Pythagorean Theorem gives:

take the square root of each side

Similarly, the other diagonal can be found with this right triangle:

Once again using Pythagorean Theorem gives an answer of

To find the length of the diagonals, split the top side into 3 sections as shown below:

The two congruent sections plus 8 adds to 14. , so the two congruent sections add to 6. They must each be 3. This means that the top of the right triangle with the diagonal as a hypotenuse must be 11, since .

We can solve for the diagonal, now pictured, using Pythagorean Theorem:

take the square root of both sides

In order to calculate the length of the diagonal, we first must assume that the height is perpendicular to both the top and bottom of the trapezoid. 

Knowing this, we can draw in the diagonal as shown below and use the Pythagorean Theorem to solve for the diagonal. 

We now take the square root of both sides: 

1) The diagonal  can be found from  by using the Pythagorean Theorem.

2) The length of the base of ,  has to be found because  is the length of the base of .

3) .

4) Using the Pythagorean Theorem on  to find ,

5) Using the Pythagorean Theorem on  to find ,

To illustrate how to determine the correct length, draw a perpendicular segment from  to , calling the point of intersection .

 divides the trapezoid into Rectangle  and right triangle  .

Opposite sides of a rectangle are congruent, so .

. The two angles of a trapezoid along the same leg - in particular,  and  - are supplementary, so 

By the 30-60-90 Triangle Theorem,

Opposite sides of a rectangle are congruent, so , and

 is the hypotenuse of right triangle , so by the Pythagorean Theorem, its length can be calculated to be

Set  and :

To illustrate how to determine the correct length, draw a perpendicular segment from  to , calling the point of intersection .

 divides the trapezoid into Rectangle  and right triangle  .

Opposite sides of a rectangle are congruent, so .

. The two angles of a trapezoid along the same leg - in particular,  and  - are supplementary, so 

By the 30-60-90 Triangle Theorem,

Opposite sides of a rectangle are congruent, so , and

 is the hypotenuse of right triangle , so by the Pythagorean Theorem, its length can be calculated to be

Set  and :

To illustrate how to determine the correct length, draw a perpendicular segment from  to , calling the point of intersection .

 divides the trapezoid into Rectangle  and right triangle  .

Opposite sides of a rectangle are congruent, so .

. The two angles of a trapezoid along the same leg - in particular,  and  - are supplementary, so 

By the 45-45-45 Triangle Theorem,

and

 is the hypotenuse of right triangle , so by the Pythagorean Theorem, its length can be calculated to be

Set  and :

A rhombus is a four-sided figure where all sides are straight and equal in length. All opposite sides are parallel. A square is considered to be a rhombus.

Solving for the area of rhombus  requires knowledge of the equation for finding the area of a rhombus. The equation is , where  and  are the two diagonals of the rhombus. Since both of these values are given to us in the original problem, we merely need to substitute these values into the equation to obtain:

The area of rhombus  is therefore  square units. 

The formula for the area of a rhombus from the diagonals is half the product of the diagonals, or in mathematical terms:

 where  and  are the lengths of the diagonals.

Substituting our values yields,