# Rhombus area - Examples, Exercises and Solutions

## How is the area of a rhombus calculated?

Every geometric problem is based on a wealth of data, whose answer is divided into several different questions. One of the most popular questions, and the one most likely to appear on your test, is the question about the area of a rhombus. As is well known, a rhombus consists of $2$ different triangles. If so, how do you calculate its area?

### Rhombus Formula

$Area=\frac{Diagonal_1\times Diagonal_2}{2}$

• The formula to calculate the area of a rhombus: Diagonal $\times$ Diagonal, divided by $2$.
• Another formula you can think of for the area of a rhombus (depending on the information provided by the question): Base $\times$ Height.

## Practice Rhombus area

### Exercise #1

Given the rhombus in the drawing:

What is the area?

### Step-by-Step Solution

Let's remember that there are two ways to calculate the area of a rhombus:

The first is the side times the height of the side.

The second is diagonal times diagonal divided by 2.

Since we are given both diagonals, we calculate it the second way:

$\frac{7\times4}{2}=\frac{28}{2}=14$

14

### Exercise #2

Look at the rhombus in the figure.

What is its area?

### Step-by-Step Solution

First, let's remember that according to the properties of a rhombus, all sides of a rhombus are equal,

Therefore, if we define the sides of the rhombus with the letters ABCD,

We can argue that:

AB=BC=CD=DA

We use the perimeter formula:

50 = AB+BC+CD+DA

And we can conclude that
4AB=50

(We can also use any other side, it doesn't matter in this case because they are all equal.)

We divide by four and reveal that:

AB=BC=CD=DA = 12.5

Now let's remember the formula for the area of a rhombus: the height times the side corresponding to the height.

We are given the length of the external height 8,

Now, we can replace in the formula:

8*12.5=100

100 cm²

### Exercise #3

Given the rhombus in the drawing:

What is the area?

### Step-by-Step Solution

Remember there are two options to calculate the area of a rhombus:

Diagonal by diagonal divided by 2.

Side by the height of the side.

In the question, we are only given half of the diagonal and the side, which means we cannot use any of the formulas.

We need to find more data. Let's find the second diagonal:

Remember that the diagonals of a rhombus are perpendicular to each other, which means they form a 90-degree angle.

Therefore, all the triangles in a rhombus are right-angled.

Now we can focus on the triangle where the side and the height are given, and we will calculate the third side by the Pythagorean theorem:

$a²+b²=c²$Replace the data:

$3^2+x^2=5^2$$9+x^2=25$$x^2=25-9=16$$x=\sqrt{16}=4$Now that we have found half of the second diagonal, we can calculate the area by diagonal by diagonal:

Since the diagonals in a rhombus are perpendicular and cross each other, they are equal. Therefore our diagonals are equal:

$3+3=6$$4+4=8$Therefore, the area of the rhombus is:

$\frac{6\times8}{2}=\frac{48}{2}=24$

24

### Exercise #4

Calculate the area of the rhombus in the figure below:

10 cm²

### Exercise #5

In the drawing given a rhombus

The length of each side of the rhombus is 5 cm

The length of the height of the side is 3 cm

What is the area of the rhombus?

15 cm².

### Exercise #1

Look at the rhombus in the diagram below.

What is the area of the rhombus?

7.5 cm²

### Exercise #2

Given the rhombus in the figure

36 cm².

### Exercise #3

Given the rhombus in the figure

80 cm²

### Exercise #4

In the drawing given a rhombus

The length of the main diagonal 6 cm

The length of the secondary diagonal 4 cm

What is the area of the rhombus?

12 cm²

### Exercise #5

Look at the rhombus in the figure.

What is its area?

42 cm²

### Exercise #1

A rhombus and its external height are shown in the figure below.

The length of each side of the rhombus is 5 cm.

What is its area?

15 cm².

### Exercise #2

Given the rhombus whose length of its sides is 8 cm

The length of the given height is 5 cm

What is the area of the rhombus?

40 cm²

### Exercise #3

Given the rhombus in the drawing:

What is the area?

52

### Exercise #4

The rhombus in the diagram has an area of 24 cm².

What is the value of X?

6

### Exercise #5

Given the rhombus in the drawing:

What is the area?