The area of a rhombus can be defined as the amount of space enclosed by a rhombus in two-dimensional space. Recall that a rhombus is a type of quadrilateral projected onto a two-dimensional (2D) plane that has four sides that are equal in length and are congruent. It is also known as an equilateral quadrilateral since its four sides are equal.

Here, we will learn how to calculate the area of a rhombus. We will learn about a formula that uses its diagonals and other formulas that use its sides. In addition, we will solve some exercises to apply the formulas.

## How to find the area of a rhombus?

To calculate the area of a rhombus, we can use one of the three main methods that exist. We can use its diagonals, we can use its base and height, and we can use trigonometry.

### Area of Rhombus using diagonals: Method 1

The area of the rhombus can be calculated using the length of its diagonals and the formula:

$latex A=\frac{d_{1}\times d_{2}}{2}$ |

where,

- $latex d_{1}=$ length of diagonl 1
- $latex d_{2}=$ length of diagonal 2
- $latex A=$ area of rhombus

**Derivation of this formula**

Consider the following rhombus:

We can represent with O the point of intersection of the two diagonals. So the area of the rhombus will be:

$latex A=4\times\text{area of }\Delta AOB$

$latex =4\times(\frac{1}{2})\times AO \times OB$

$latex =4\times(\frac{1}{2})\times(\frac{1}{2})d_{1}\times(\frac{1}{2})d_{2}$

$latex =4\times(\frac{1}{8})d_{1}d_{2}$

$latex =\frac{1}{2}d_{1}d_{2}$

Therefore, the area of the rhombus is $latex A=\frac{1}{2}d_{1}d_{2}$, where, $latex d_{1}$ and $latex d_{2}$ are the diagonal of a rhombus.

Thus, considering a rhombus ABCD that has two diagonals, for example, AC and BD, we can use the following steps to calculate its area:

**Step 1:** Find the length of diagonal 1, $latex d_{1}$. The diagonals of a rhombus are perpendicular to each other forming four right angles when they intersect.

** Step 2:** Find the length of diagonal 2, $latex d_{2}$.

** Step 3:** Multiply both diagonals, $latex d_{1}$ and $latex d_{2}$.

** Step 4:** Divide the result by 2.

### EXAMPLE 1

Find the area of a rhombus that has diagonals equal to 8 cm and 10 cm.

**Solution:** We have the following information:

- Diagonal 1, $latex d_{1}=8$ cm
- Diagonal 2, $latex d_{2}=10$ cm

We use the formula for the area of the rhombus with the given information:

$latex A=\frac{d_{1}\times d_{2}}{2}$

$latex =\frac{8\times 10}{2}$

$latex =\frac{80}{2}$

$latex A=40$

Therefore, the area of the rhombus is 40 cm².

### EXAMPLE 2

What is the area of a rhombus that has diagonals equal to 10 m and 12 m?

**Solution:** We have the following values:

- Diagonal 1, $latex d_{1}=10$ m
- Diagonal 2, $latex d_{2}=12$ m

Replacing these values in the formula, we have:

$latex A=\frac{d_{1}\times d_{2}}{2}$

$latex =\frac{10\times 12}{2}$

$latex =\frac{120}{2}$

$latex A=60$

Therefore, the area of the rhombus is 60 m².

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## How to calculate the area of a rhombus without diagonals?

The area of a rhombus can also be calculated without the diagonals. We have two methods to do this.

### Area of Rhombus using base and height: Method 2

The area of the rhombus can be calculated using the base and the height with the formula:

$latex A=bh$ |

where,

- $latex b=$ length of either side of the rhombus
- $latex h=$ length of the rhombus height
- $latex A=$ area of the rhombus

Therefore, we calculate the area of the rhombus from its base and height with the following steps:

**Step 1:** Find the height and base of the rhombus. The base of the rhombus is one of its sides and the height is the perpendicular distance from the chosen base to the opposite side.

**Step 2:** Multiply the base and the height obtained.

### EXAMPLE

What is the area of a rhombus that has a base of 8 m and a height of 6 m?

**Solution:** We have the following values:

- Base, $latex b=8$ m
- Height, $latex h=6$ m

Using these values with the formula, we have:

$latex A=bh$

$latex =(8)(6)$

$latex A=48$

Thus, the area of the rhombus is 48 m².

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### Area of Rhombus using trigonometry: Method 3

We can calculate the area of a rhombus using trigonometry with the following formula:

$latex A={{b}^2}\times \sin(a)$ |

where,

- $latex b=$ length of either side of the rhombus
- $latex a=$ measure of any internal angle
- $latex A=$ area of the rhombus

Therefore, we can use the following steps to calculate the area of a rhombus using trigonometry:

**Step 1:** Square the length of either side of the rhombus.

**Step 2:** Multiply the result from step 1 by the sine of one of its angles.

### EXAMPLE

A rhombus has sides with a length of 10 m and an internal angle of 60°. What is its area?

**Solution:** We have the following data:

Side, $latex b=10$ m

Angle, $latex a=60$°

Using this information in the formula, we have:

$latex A={{b}^2}\times \sin(60°)$

$latex ={{10}^2}\times 0.866$

$latex =100\times 0.866$

$latex A=866$

Therefore, the area of the rhombus is 866 m².

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## See also

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