# Rhombus

In Euclidean geometry, a rhombus is a type of quadrilateral. It is a special case of a parallelogram, whose diagonals intersects each other at 90 degrees. This is the basic property of rhombus. The shape of a rhombus is in a diamond shape, hence it is also called a diamond. You must have seen the diamond shape in the playing cards. All the rhombi are parallelogram and kite. And if the angles of the rhombus are all 90 degrees, then it is a square.

Now, before we discuss more about rhombus and its properties, Â let us know what is a quadrilateral? A quadrilateral is a polygon containing 4 sides and 4 vertices enclosing 4 angles. The sum of the interior angles of a quadrilateral is equal to 360 degrees. The quadrilateral is basically of 6 types such as:

1. Parallelogram
2. Trapezium
3. Square
4. Rectangle
5. Kite
6. Rhombus

## Rhombus Definition

A rhombus is a special case of a parallelogram and it is a four-sided quadrilateral. In a rhombus, opposite sides are parallel and opposite angles are equal. Moreover, all the sides of a rhombus are equal in length and the diagonals bisect each other at right angles. The rhombus is also called a diamond or rhombus diamond. The plural form of a rhombus is rhombi or rhombuses.

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

### Is square a Rhombus?

Rhombus has all its sides equal and so do the square. Also, the diagonals of the square are perpendicular to each other and bisect the opposite angles. Therefore, a square is a type of rhombus.

### Angles of Rhombus

The opposite angles of a rhombus are equal to each other. Also, the diagonals of a rhombus bisect these angles.

## Rhombus Formulas

The formulas for rhombus are defined for two major attributes such as:

1. Area
2. Perimeter

### Area of Rhombus

The area of the rhombus is the region covered by it in a two-dimensional plane. The formula for the area is equal to the product of diagonals of rhombus divided by 2. It can be represented as:

 Area of Rhombus, A = (d1 x d2)/2 square units

where d1 and d2 are the diagonals of a rhombus.

### Perimeter of Rhombus

The perimeter of a rhombus is the total length of its boundaries. Or we can say the sum of all the four sides of a rhombus is its perimeter. The formula for its perimeter is given by:

 The perimeter of Rhombus, P = 4a units

Where the diagonals of the rhombus are d1Â & d2Â and ‘a’ is the side.

## Properties of Rhombus

Some of the important properties of the rhombus are as follows:

• All sides of the rhombus are equal.
• The opposite sides of a rhombus are parallel.
• Opposite angles of a rhombus are equal.
• In a rhombus, diagonals bisecting each other at right angles.
• Diagonals bisect the angles of a rhombus.
• The sum of two adjacent angles is equal to 180 degrees.
• The two diagonals of a rhombus form four right-angled triangles which are congruent to each other
• You will get a rectangle when you join the midpoint of the sides.
• You will get another rhombus when you join the midpoints of half the diagonal.
• Around a rhombus, there can be no circumscribing circle.
• Within a rhombus, there can be no inscribing circle.
• You will get a rectangle, where the midpoints of the 4 sides are joined together and the length and width of the rectangle will be half the value of the main diagonal. So that the area of the rectangle will be half of the rhombus.
• When the shorter diagonal is equal to one of the sides of a rhombus, two congruent equilateral triangles are formed.
• You will get a cylindrical surface having a convex cone at one end and concave cone at another end when the rhombus is revolved about any side as the axis of rotation
• You will get a cylindrical surface having concave cones on both the ends when the rhombus is revolved about the line joining the midpoints of the opposite sides as the axis of rotation
• You will get solid with two cones attached to their bases when the rhombus is revolving about the longer diagonal as the axis of rotation. In this case, the maximum diameter of the solid is equal to the shorter diagonal of the rhombus.
• You will get solid with two cones attached to their bases when the rhombus is revolving about the shorter diagonal as the axis of rotation. In this case, the maximum diameter of the solid is equal to the longer diagonal of the rhombus.

## Rhombus Solved Problems

The sample example for the rhombus is given below.

Question:

The two diagonal lengths d1 and d2 of a rhombus are 6cm and 12 cm respectively. Find its area.

Solution:

Given:

Diagonal d1 = 6cm

Diagonal d2= 12 cm

Area of the rhombus, A = (d1 x d2)/2 square units

A = ( 6 x 12)/2

A = 72/2

A = 36 cm2

Therefore, the area of rhombus = 36 square units.

Question 2:

Find the diagonal of a rhombus if its area is 121 cm2 and length measure of longest diagonal is 22 cm.

Solution:Â

Given: Area of rhombus = 121 cm2 and Lets say d1 = 22 cm.

Using Area of the rhombus formula,Â  A = (d1 x d2)/2 square units, we get

121 = (22 x d2)/2

121 = 11 x d2

orÂ  11 =Â  d2

Question 3:

What are the basic properties of rhombus?

Solution:

The basic properties of the rhombus are:

1. The opposite angles are congruent.
2. The diagonals intersect each other at 90 degrees.
3. The diagonals bisect the opposite interior angles.
4. The adjacent angles are supplementary.

Question 4:

What is the perimeter of a rhombus whose sides are all equal to 6 cm?

Solution:

Given, the side of rhombus = 6cm

Since all the sides are equal, therefore,

Perimeter = 4 x side

P = 4 x 6

P = 24cm

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