# Advanced Geometry : How to find the area of a rhombus

## Example Questions

### Example Question #11 : How To Find The Area Of A Rhombus

Find the area of a rhombus with diagonal lengths of  and .

Explanation:

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:

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

### Example Question #12 : How To Find The Area Of A Rhombus

The above figure shows a rhombus . Give its area.

Explanation:

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

.

### Example Question #13 : How To Find The Area Of A Rhombus

Rhombus  has perimeter 48; . What is the area of Rhombus  ?

Explanation:

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:

### Example Question #14 : How To Find The Area Of A Rhombus

A rhombus contains diagonals with the length  and . Find the area of the rhombus.

Explanation:

The equation for the area of a rhombus is given by:

where  and  are the two diagonal lengths.

This problem very quickly becomes one of the "plug and chug" type, where the given values just need to be substituted into the equation and the equation then solved. By plugging in the values given, we get:

### Example Question #15 : How To Find The Area Of A Rhombus

Find the area of a rhombus if the diagonals lengths are  and .

Explanation:

Write the formula for the area of a rhombus:

Substitute the given lengths of the diagonals and solve:

### Example Question #16 : How To Find The Area Of A Rhombus

Find the area of a rhombus if the diagonals lengths are  and .

Explanation:

Write the formula for finding the area of a rhombus. Substitute the diagonals and evaluate.

### Example Question #242 : Geometry

A rhombus has a side length of 5. Which of the following is NOT a possible value for its area?

24

25

10

30

15

30

Explanation:

The area of a rhombus will vary as the angles made by its sides change. The "flatter" the rhombus is (with two very small angles and two very large angles, say 2, 178, 2, and 178 degrees), the smaller the area is. There is, of course, a lower bound of zero for the area, but the area can get arbitrarily small. This implies that the correct answer would be the largest choice. In fact, the largest area of a rhombus occurs when all four angles are equal, i.e. when the rhombus is a square. The area of a square of side length 5 is 25, so any value bigger than 25 is impossible to acheive.

### Example Question #18 : How To Find The Area Of A Rhombus

Find the area of the rhombus shown below. You will have to find the lengths of the sides as well.

The rhombus shown has the following coordinates:

Round to the nearest hundredth.

Explanation:

Finding the area of a rhombus follows the formula:

In this rhombus, you will find that , which are the two x coordinates.

The length of q is a more involved process. You can find q by using the Pythagorean Theorem.

.

Therefore, the area is

.

### Example Question #11 : Rhombuses

Show algebraically how the formula for the area of a rhombus is developed.

Explanation:

1) The given rhombus is divded into two congruent isosceles triangles.

2) Each isosceles triangle has a height  and a base .

3) The area  of each isosceles triangle is .

4) The areas of the two isosceles triangles are added together,

### Example Question #20 : How To Find The Area Of A Rhombus

Find the area of the rhombus below.

Explanation:

Recall that the diagonals of the rhombus are perpendicular bisectors. From the given side and the given diagonal, we can find the length of the second diagonal by using the Pythagorean Theorem.

Let the given diagonal be diagonal 1, and rearrange the equation to solve for diagonal 2.

Plug in the given side and diagonal to find the length of diagonal 2.

Now, recall how to find the area of a rhombus:

Plug in the two diagonals to find the area.