# Properties of a Rhombus

In geometry, a rhombus is a type of quadrilateral. The rhombus is a special case of a parallelogram that has diagonals that intersect each other at 90 degrees. This is the basic property of a rhombus. Rhombuses are shaped like a diamond, which is why they are often called diamonds. If all the interior angles of a rhombus are 90 degrees, then the rhombus is a square.

Here, we will learn about the fundamental properties of rhombuses. Also, we will look at the important rhombus formulas and some examples of rhombus problems that illustrate the application of these formulas.

##### GEOMETRY

Relevant for

Learning about the fundamental properties of rhombuses.

See properties

##### GEOMETRY

Relevant for

Learning about the fundamental properties of rhombuses.

See properties

## Definition of a rhombus

A rhombus is a special case of the parallelogram and is a quadrilateral with four equal sides. In a rhombus, the opposite sides are parallel and the opposite angles are equal.

Also, all the sides of the rhombus are equal in length and the diagonals bisect each other at right angles. The rhombus is also known as a diamond.

In the figure above, we can see the rhombus ABCD, where AB, BC, CD, and AD are the sides of the rhombus and AC and BD are the diagonals of the rhombus.

Is the square a rhombus?

A rhombus has all its sides of equal length and a square also does. Moreover, the diagonals of the square are perpendicular to each other and bisect at opposite angles. Therefore, a square is a type of rhombus.

## Fundamental properties of the rhombus

Some of the important properties of rhombuses are as follows:

• All sides of the rhombus are equal.
• The opposite sides of a rhombus are parallel.
• The opposite angles of a rhombus are equal.
• In a rhombus, the diagonals bisect each other at right angles.
• The diagonals bisect the interior angles of the rhombus.
• The sum of the adjacent angles is equal to 180 degrees.
• The two diagonals of a rhombus form four right triangles that are congruent.
• There cannot be a circumscribed circle around a rhombus.
• There cannot be a circle inscribed within a rhombus.
• If we join the midpoints of the sides, we will obtain a rectangle.
• When the shortest diagonal is equal to one of the sides of the rhombus, two congruent equilateral triangles are formed.

## Important rhombus formulas

The following formulas are useful for solving problems involving rhombuses.

### Area of a Rhombus

The area of the rhombus is the region covered by the rhombus in the two-dimensional plane. The formula for the area is equal to the product of the diagonals of the rhombus divided by 2:

where, $latex d_{1}, ~d_{2}$ are the diagonals of a rhombus.

### Perimeter of the rhombus

The perimeter of the rhombus is the total length of its boundaries. The perimeter is equal to the sum of the lengths of the four sides of the rhombus:

where l is the length of one of the sides of the rhombus.

## Examples of rhombus problems

### EXAMPLE 1

A rhombus has diagonals of length 12 m and 10 m. What is its area?

Solution: We have the following information:

• Diagonal 1: $latex d_{1}=12$ m
• Diagonal 2: $latex d_{2}=10$ m

Therefore, we can use this information in the area formula:

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

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

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

$latex A=60$

The area of the rhombus is 60 m².

### EXAMPLE 2

A rhombus has sides of length 15 m. What is its perimeter?

Solution: We have that the length of the sides of the rhombus is 15 m. Therefore, we use the perimeter formula with this value:

$latex p=4l$

$latex =4(15)$

$latex =60$

$latex p=60$

The perimeter of the rhombus is 60 m.  