**Rhombus Definition**

Rhombus is a quadrilateral in which **all sides are equal **with **parallel opposite sides** and **equal opposite angles**

**Keyword for Rhombus**

(a) All equal sides

(b) Opposite sides parallel

(c) Opposite angle equal

**Structure of Rhombus**

Important elements of Rhombus are**(a) Sides**

Rhombus has four sides of equal length

AB = BC = CD = DA

**(b) Angles**

It has four angles

In Rhombus, opposite angles are equal

i.e. ∠A = ∠C and ∠D = ∠B

**(c) Vertex**

The point where two sides join is called vertex.

Rhombus has 4 vertex

Point A, B, C & D are the vertex in above figure

**(d) Shape**

Rhombus looks like a diamond or a star

**Properties of Rhombus**

**(01) All sides are equal**

In the above Rhombus ABCD;

AB = BC = CD = DA

All the sides have equal length

**(02) Opposite sides are parallel**

Observe the above image in property 01

Here side AB is parallel to CD and side DA is parallel to CB**AB || CD and DA || CB**

**(03) In Rhombus, opposite angles are equal**

In the above image of Rhombus:

∠A = ∠C = 100 degree

∠B = ∠D = 80 degree

Both set of opposite angles are equal

**(04) In Rhombus adjacent angles are supplementary**

It means that sum of adjacent angles will add to 180 degree

∠A + ∠B = 180 degree

∠B + ∠C = 180 degree

∠C + ∠D = 180 degree

∠D + ∠A = 180 degree

**(05) Diagonals of Rhombus bisect each other at right angle**

The property is similar to square

Given above is the Rhombus ABCD with diagonals AC and BD

The diagonals bisect each other. This means that diagonals divide each other into equal halves.

i.e. **AO = OC & DO = OB**

Also diagonals intersect each other at right angles

**(06) Diagonals also divides the angle into equal halves**

Given above is the rhombus ABCD with diagonal AC

The diagonal divides the angle A & C into two equal halves.

Hence,

∠1 = ∠2

∠3 = ∠4

**(07) Perimeter of Rhombus**

Perimeter is the length of total boundary of any geometrical figure

Formula for Perimeter of Rhombus is:

Perimeter = Side + Side + Side + Side**Perimeter = 4 x Side**

**(08) Area of Rhombus**

**Area of Rhombus = (1/2) x d1 x d2**

Where d1 & d2 are the lengths of the diagonal

**(09) Making rectangle inside the rhombus**

On joining the midpoint of the side of Rhombus, you will get a rectangle

ABCD is a Rhombus and P, Q, R & S are the midpoint of the respective points.

On joining the points P, Q, R and Q we get a rectangle

**(10) Rhombus inside Rhombus**

If you join the midpoint of half the diagonal you will get another Rhombus

ABCD is a Rhombus and AC and BD are its diagonals

AO is half of diagonal AC & P is midpoint of AO;

Similarly, R is the midpoint of OC;

Q & S are midpoints of OB & OD respectively;

**On joining all the points P, Q, R & S you get another Rhombus PQRS**

**Frequently Asked Questions – Rhombus **

**(01) Are all squares a form of Rhombus?**

Yes!!

In every square

(a) All sides are equal

(b) Opposite sides are parallel

(c) Opposite angles are equal

Hence, all squares can be called Rhombus

**(02) Are all Rhombus a form of square?**

NO!!

In Rhombus each angles may or may not be 90 degree.

Hence, all Rhombus doesn’t fulfill the requirement of Square

**(03) Are all angles in a Rhombus equal?**

NO!!

In a Rhombus, only opposite angles are equal

**(04) What are the main difference between Rhombus and Parallelogram?**

In Rhombus all sides are equal.

In Parallelogram, only opposite sides are equal

Rhombus looks like a star while parallelogram looks completely different

**Solved Questions – Rhombus**

**(01) The length of two diagonal of Rhombus are 4 cm and 6 cm. Find the area of Rhombus**

**Solution**

We know that:

Area of Rhombus = (1/2) x d1 x d2

Putting the value of both the diagonals in the formula

Area of Rhombus = (1/2) x 4 x 6 = 12 sq. cm

**Hence, 12 sq. cm is the required area**

**(02) Find the perimeter of rhombus of side 5 cm **

**Solution**

Perimeter of Rhombus = 4 x (side)

Putting the value of side length ,we get;

Perimeter of Rhombus = 4 x 5 = 20 cm

**Hence, 20 cm is the required perimeter of the Rhombus**

**(03) If the area of Rhombus is 40 sq. cm and length of one diagonal is 8 cm. Find the length of other diagonal**

**Solution**

Given

Area of Rhombus = 40 sq. cm

Diagonal length (d1) = 8 cm

To find

Diagonal Length (d2) = ?

We know that:

Area of Rhombus = 1/2 x d1 x d2

Putting the values in the formula, we get;

40 = (1/2) x 8 x d2

80 = 8 x d2

d2 = 80/8

d2 = 10 cm

**Hence, length of other diagonal is 10 cm **

**(04) Given below is the Rhombus ABCD, where ∠A = 77 degreeFind the measurement of ∠B**

In Rhombus, alternate angles are supplementary**∠A** + **∠B** **= 180****77 **+ **∠B** **= 180****∠B** **= 180 – 77**

**∠B** **= 103 degree**

**Hence ∠B measures 103 degree**