# Rhombuses

By this point, you may have come across quadrilaterals quite a lot. If not, know that a quadrilateral is any four-sided shape. You may also remember learning about the different types of quadrilaterals. Of course, we have the square and the rectangle. But we also have those others that are less common and can be harder to remember, such as parallelograms, trapezoids, and kites. In this article, we discuss rhombuses. Let's start by talking about what we mean by a rhombus

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## Rhombuses meaning

• A rhombus is a quadrilateral with all sides equal in length and two pairs of parallel sides.
• A quadrilateral is any four-sided shape.
• A parallelogram is any quadrilateral with two pairs of parallel sides.

A quadrilateral with 2 pairs of parallel opposite sides is called a parallelogram.

From the definition of the rhombus and parallelogram, we see that all rhombuses are special types of parallelograms, since they have two pairs of parallel sides. It is helpful to be familiar with parallelograms for the purposes of this article.

The following figure illustrates a rhombus, $ABCD$. Since it is a rhombus, all of its sides are of equal length.

Rhombus illustration - StudySmarter Originals

From the figure of ABCD, we have:

$AB=BC=CD=DA$

Since a rhombus is a type of parallelogram, the opposite sides are parallel. Thus, $AD$ is parallel to $BC$ and $AB$is parallel to $BC$.

## Properties of rhombuses

Now that we have discussed the basic characteristics of a rhombus, let's consider its properties in more detail.

### Rhombuses as parallelograms

We have mentioned that rhombuses are a special type of parallelogram, so we can say that all the properties of parallelograms apply to rhombuses as well. Let's see how the properties of parallelograms in general apply specifically to rhombuses:

• Both pairs of opposite sides of a rhombus are parallel.

• Opposite angles of a rhombus are equal.

• The diagonals of a rhombus bisect each other. In other words, the intersection of the two diagonals is at the mid point of each diagonal.

• Each diagonal of a rhombus divides the rhombus into 2 congruent triangles.

### Properties unique to rhombuses

In addition to the properties concerning parallelograms, there are also additional properties specific and unique to rhombuses. We will describe this with reference to the rhombus ABCD below:

Rhombus illustration with diagonals shown - StudySmarter Originals
• The diagonals of a rhombus are perpendicular to each other. This means they are at right angles to each other.
• Thus, in the diagram above, $AC\perp BD$.
• From this, we could also say that $\angle AOD=\angle COD=\angle COB=\angle AOB=90°$
• Each diagonal of a rhombus bisects a pair of opposite interior angles.
• In other words, $\angle CAB=\angle CAD=\angle ACB=\angle ACD$ and $\angle ABD=\angle CBD=\angle ADB=\angle CDB$
• The four triangles created when we add in the rhombus diagonals are congruent. So, they are mathematically identical but just oriented differently.

### The congruent triangles of rhombuses

From the properties of rhombuses, we know that its diagonals divide the shape into four triangles that are congruent. What does it mean for triangles to be congruent? Two or more triangles are congruent if they are mathematically identical. In other words, all of the sides and angles are the same, even if they are oriented differently. Also recall that the internal angles in a triangle sum to 180 degrees.

Consider the rhombus below. Prove that for the given rhombus, AC ⊥ BD.

The mathematical symbol means "perpendicular to."

Rhombus example - StudySmarter Originals

Solution:

By the definition of a rhombus:

$AB=BC=CD=DA$

Now consider the triangles $AOB$ and $COB$:

The side $OB$ is a side of both triangles. Now, $AB=BC$, since $ABCD$ is a rhombus. We also have $OA=OC$, since $ABCD$ is also a parallelogram and the diagonals of a parallelogram bisect each other. Therefore, triangle $AOB$ is congruent to triangle $COB$. In other words, they are exactly the same triangles, just rotated in different positions.

This implies that:

$\angle AOB=\angle COB$

Additionally, $\angle AOB$ and $\angle COB$ lie on the same straight line. Thus,

$\angle AOB+\angle COB=180°$

Therefore,

$\angle AOB=\angle COB=180÷2=90°$

Similarly we can show that:

$\angle AOD=\angle COD=90°$

Thus, $AC$ and $BD$ are perpendicular. That is, $AC\perp BD$.

## Area formula for rhombuses

We have a specific formula to find the area of a rhombus. Consider the following rhombus:

Rhombus example - StudySmarter Originals

Now, let's label the diagonals such that $BD={d}_{1}$ and $AC={d}_{2}$.

The area of the rhombus is given by the formula:

$Area\mathit{}\mathit{=}\mathit{}\frac{\mathit{1}}{\mathit{2}}{d}_{\mathit{1}}{d}_{\mathit{2}}$

Since a rhombus is a type of quadrilateral, we also have an alternative formula for finding the area.

The area of any quadrilateral is given by the formula:

$Area\mathit{}\mathit{=}\mathit{}base\mathit{}\mathit{×}\mathit{}height$

So, depending on what lengths we have, we can use either of the above formulas to work out the area of a rhombus.

Note that when we discuss area, we use square units. For example, if the lengths of the base and height are given both in centimeters, the units for the area are centimeters squared ($c{m}^{2}$).

A rhombus has diagonals of lengths $10cm$ and $15cm$. What is the area of the rhombus?

Solution:

Using the specific formula for the area of a rhombus, we have:

$Area\mathit{}\mathit{=}\mathit{}\frac{\mathit{1}}{\mathit{2}}{d}_{\mathit{1}}{d}_{\mathit{2}}$

Substituting in ${d}_{1}=10$and ${d}_{2}=15$, we have:

$\mathrm{Area}=\frac{1}{2}×10cm×15cm\phantom{\rule{0ex}{0ex}}=75c{m}^{2}$

Thus, the area of this rhombus is $75c{m}^{2}$.

## Further rhombuses examples

Now we will look at some further example problems on rhombuses.

Consider the below rhombus. Given that $\angle ACB=35°$, find $\angle DBA$.

Rhombus example - StudySmarter Originals

Solution:

Recall that in a rhombus, each diagonal bisects, which means that we have a pair of equal angles. We also know that opposite angles in a rhombus are equal.

Thus,

$\angle ACB=\angle OAB=35°$

Now, we mentioned earlier that the diagonals in a rhombus are perpendicular to each other. Therefore,

$\angle AOB=90°$

Since $OAB$ forms a triangle, and angles in a triangle sum to $180°$, we can work out $\angle DBA$:

$\angle DBA+\angle AOB+\angle OAB=180°$

So, substituting in the known angles, we have

$\angle DBA+90°+35°=180°$

which implies that

$\angle DBA+125°=180°$

Subtracting $180°$ from both sides, we obtain:

$\angle DBA=180°-125°=55°$

So, we have that $\angle DBA=55°$.

Consider the following rhombus depicted below. Given that $\angle ADC=120°$, find $\angle OAD$.

Rhombus example - StudySmarter Originals

Solution:

Recall that diagonal $BD$ bisects $\angle ADC$. Therefore, we have,

$\angle ADO=120°÷2=60°$.

We also have that $\angle DOA=90°$ due to the perpendicular bisector.

So, since angles in a triangle sum to $180°$, we have:

$\angle OAD=180°-90°-60°=30°$.

Thus, $\angle OAD=30°$.

## Rhombuses - Key takeaways

• A rhombus is a special quadrilateral with all four sides equal in length and two pairs of parallel sides.
• A parallelogram is any quadrilateral with two pairs of parallel sides, so a rhombus is also a parallelogram.
• Opposite angles of a rhombus are equal.
• The diagonals of a rhombus bisect each other. In other words, the intersection of the two diagonals is at the midpoint of each diagonal.
• The diagonals of a rhombus divide the shape into four congruent right-angled triangles.
• The area of a rhombus is given by $\frac{\mathit{1}}{\mathit{2}}{d}_{\mathit{1}}{d}_{\mathit{2}}$ where ${d}_{1}$ and ${d}_{2}$ are the diagonals.

#### Flashcards inRhombuses 14

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What are rhombuses?

A rhombus is a quadrilateral with all 4 of its sides equal.

What shape is a rhombus?

A rhombus is a quadrilateral with all 4 of its sides equal.

Is a square a rhombus?

A square is a special type of rhombus.

How to find the area of a rhombus?

Consider a rhombus with diagonals of length d1 and d2. The area of the rhombus is given by: A = 0.5×d1×d2.

What are the properties of rhombuses?

• The diagonals of a rhombus are perpendicular to each other.
• If the diagonals of a parallelogram are perpendicular, then it implies that the given parallelogram is a rhombus.
• Each diagonal of a rhombus bisects a pair of opposite angles.

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