Rhombus

In Euclidean geometry, a rhombus is a type of quadrilateral. It is a special case of a parallelogram, whose all sides are equal and diagonals intersect 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. Check lines of symmetry in a rhombus.

You must have seen the diamond shape in the playing cards. All the rhombi are parallelograms and kites. If all the angles of the rhombus are 90 degrees, then it is a square.

Now, before we discuss rhombus and its properties,  let us know what a quadrilateral is? A quadrilateral is a closed 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. In a rhombus, opposite sides are parallel and the 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 a rhombus and AC and BD are the diagonals of a rhombus.

Is Square a Rhombus?

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

Angles of Rhombus

Below are some important facts about the rhombus angles:

• Rhombus has four interior angles.
• The sum of interior angles of a rhombus adds up to 360 degrees.
• The opposite angles of a rhombus are equal to each other. 
• The adjacent angles are supplementary.
• In a rhombus, diagonals bisect each other at right angles.
• 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

Get more: Mathematics formulas

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 the rhombus divided by 2. It can be represented as:

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:

Where the diagonals of the rhombus are d₁ & d₂ 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 bisect each other at right angles.
• Diagonals bisect the angles of a rhombus.
• The sum of two adjacent angles is equal to 180 degrees.
• 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 inscribed 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 1:

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 cm²

Question 2:

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

Solution: 

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

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

121 = (22 x d₂)/2

121 = 11 x d₂

or  11 =  d₂

Therefore, the Length of another diagonal is 11 cm.

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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