One of the formulas for a rhombus is base times height,

Since a rhombas has equal sides, the base is 5 and the height of the rhombus is the same as the height of the rectangle, 4.

Substituting in these values we get the following:

To solve for the area of the rhombus , we must use the equation , where  and  are the diagonals of the rhombus. Since the perimeter of the rhombus is , and by definition all 4 sides of a rhombus have the same length, we know that the length of each side is . We can find the length of the other diagonal if we recognize that the two diagonals combined with a side edge form a right triangle. The length of the hypotenuse is , and each leg of the triangle is equal to one-half of each diagonal. We can therefore set up an equation involving Pythagorean's Theorem as follows:

, where  is equal to one-half the length of the unknown diagonal.

We can therefore solve for  as follows:

 is therefore equal to 8, and our other diagonal is 16. We can now use both diagonals to solve for the area of the rhombus:

The area of rhombus  is therefore equal to 

The area of a rhombus is given below. Plug in the diagonals and solve for the area.

The area of the rhombus is given below.  Substitute the diagonals into the formula.

Write the equation for the area of a rhombus.

Substitute the diagonals and evaluate the area.

Write the formula for the area of a rhombus.

Since both diagonals are equal, .  Plug in the diagonals and reduce.

Write the formula for an area of a rhombus.

Substitute the diagonal lengths provided into the formula.

Multiply the two terms in the numerator.

You can consider the outermost division by two as multiplying everything in the numerator by .

Multiply across and reduce to arrive at the correct answer.

Write the formula for the area of a rhombus.

Substitute the given diagonal lengths:

Use FOIL to multiply the two parentheticals in the numerator:

First: 

Outer: 

Inner: 

Last: 

Add your results together:

Divide all elements in the numerator by two to arrive at the correct answer:

Construct the other diagonal of the rhombus, which, along with the first one, form a pair of mutual perpendicular bisectors.

By the Pythagorean Theorem, 

The rhombus can be seen as the composite of four congruent right triangles, each with legs 10 and , so the area of the rhombus is 

.

Each side of a rhombus is congruent, so if it has perimeter 48, it has sidelength 12. Also, the diagonals of a rhombus are each other's perpendicular bisectors, so if they are both constructed, and their point of intersection is called , then . The following figure is formed by the rhombus and its diagonals.

 is a right triangle with its short leg half the length of its hypotenuse, so it is a 30-60-90 triangle, and its long leg measures  by the 30-60-90 Theorem. Therefore, . The area of a rhombus is half the product of the lengths of its diagonals: