# What is a Rhombus

By the end of this lesson, you will be able to

- define rhombus.
- calculate the area of a rhombus shape.
- determine the perimeter of a rhombus.
- interpret properties of rhombus.
- derive the diagonal of rhombus formula.
- relate rhombus to parallelogram.
- analyze the relationship between sides and angles of a rhombus.
- demonstrate examples of scalene triangle.

## Rhombus

A rhombus is a convex quadrilateral with the four sides are of equal length.

Rhombus is a special type of quadrilateral.

In fact, it is a special type of parallelogram.

It is sometimes called diamond because it looks like a diamond.

Again it is an equilateral quadrilateral because all four sides are the same length.

### Rhombus Example

## Rhombus Formula

- perimeter of rhombus formula
- diagonal of rhombus formula
- length of sides of rhombus formula
- area of rhombus formula

## Perimeter of Rhombus

Sum of the length of all sides of a rhombus is called the perimeter of rhombus.

Length of the sides are equal of a rhombus. So, one can find the perimeter of a rhombus if the length of the side is given. Again, length of side can be determined if the length of the two diagonal are given. And then one can easily evaluate the perimeter denpending on the side length.

### Perimeter of Rhombus Formula

Let ABCD be a rhombus and AB = BC = CD = AD = a.

It is known to us, length of sides are equal.

If the perimeter is P,

P = (AB + BC + CD + AD) unit

or, P = (a + a + a + a) unit

∴ P = 4a unit

A rhombus is showing its sides.

If the length of the side of a rhombus ABCD is a and the perimeter is P,

P = 4a unit

## Diagonal of Rhombus

The length of the line segment connecting two opposite vertices of a rhombus is called the diagonal of rhombus.

There are two pair of opposite vertices in a rhombus. So, the two pair of opposite vertices produce two diagonals. Hence, a rhombus contains two diagonals.

Moreover, two diagonals of a rhombus intersects at a right angle to each other.

### Diagonal of Rhombus Formula

Length of the diagonal can be calculated depending upon length of the side. Let's go how to derive the formula for the diagonal of rhombus.

Let the length of the side of a rhombus ABCD be AB, BC, CD, AD; and its two diagonals be AC and BD.

As the sides of a rhombus are equal to one another, let AB = BC = CD = AD = a and its two diagonals AC = d_{1} and BD = d_{2}

A rhombus is showing its diagonals.

Applying law of cosines) in △ABC,

AC^{2} = AB^{2} + BC^{2} - 2AB.BC.cosB

or, d_{1}^{2} = a^{2} + a^{2} - 2a.a cosB

or, d_{1} = √2a^{2} - 2a^{2} cosB

or, d_{1} = √2a^{2} - 2a^{2} cos(180°-A) [∵ A + B = 180°]

or, d_{1} = √2a^{2} - 2a^{2} (-cosA)

or, d_{1} = √2a^{2} + 2a^{2} cosA

or, d_{1} = √a^{2}(2 + 2cosA)

∴ d_{1} = a√2 + 2cosA

Again applying law of cosines) in △ABD,

BD^{2} = AB^{2} + AD^{2} - 2AB.AD.cosA

or, BD = √AB^{2} + AD^{2} - 2AB.AD.cosA

or, d_{2} = √AB^{2} + AD^{2} - 2a.a cosA

or, d_{2} = √a^{2} + a^{2} - 2a^{2}cosA

or, d_{2} = √2a^{2} - 2a^{2}cosA

or, d_{2} = √a^{2}(2 - 2cosA)

or, d_{2} = a√2 - 2cosA

If the length of the side of a rhombus ABCD is a and the large diagonal and small diagonal are d_{1} and d_{2} respectively,

d_{1} = a√2 + 2cosA unit

d_{2} = a√2 - 2cosA unit

## Sides of Rhombus Formula

The length of the side of rhombus can be determined if the diagonals are known. Again, the length of the side also found if area and a diagonal of a rhombus are given. Let's go how to derive a formula for finding the length of the side depending on the diagonals.

Fisrt Method: Suppose, ABCD is a rhombus of side length a whose two diagonals are d_{1} ও d_{2}. the diagonals d_{1} ও d_{2} intersects at the point O. It is known that the diagonals of rhombus meet at a right angle and they bisect themselves.

∴ OA = d_{1}2 and OB = d_{2}2

∴ OA^{2} = d_{1}^{2}4 and OB^{2} = d_{2}^{2}4

Again, ∠AOB = 90°

∴ Applying pythagorean theorem from right △AOB, it can be written as

AB^{2} = OA^{2} + OB^{2}

or, a^{2} = d_{1}^{2}4 + d_{2}^{2}4

or, a^{2} = d_{1}^{2} + d_{2}^{2}4

or, a = √d_{1}^{2} + d_{2}^{2}√4

∴ a = √d_{1}^{2} + d_{2}^{2}2

Second Method: It is known to us, sum of the area of squares drawn on the sides of a parallelogram is equal to the sum of the area of squares drawn on its two diagonals. A rhombus is also a parallelogram. Therefore, applying this on rhombus, it can be written as,

∴ AB^{2} + BC^{2} + CD^{2} + AD^{2} = d_{1}^{2} + d_{2}^{2}

or, a^{2} + a^{2} + a^{2} + a^{2} = d_{1}^{2} + d_{2}^{2}

or, 4a^{2} = d_{1}^{2} + d_{2}^{2}

or, a = √d_{1}^{2} + d_{2}^{2}√4

∴ a = √d_{1}^{2} + d_{2}^{2}2

If the length of the sides of a rhombus ABCD is a and two diagonals are d_{1} and d_{2},

a = √d_{1}^{2} + d_{2}^{2}2

## Area of Rhombus

Area of a rhombus means how much surface it contains by its four sides. Area of a rhombus can be determined if the diagonals are known.

Again, the diagonal of a rhombus can be calculated if the length of one side and its adjacent angle are given. And then, one can find the area of a rhombus depending on its diagonal. The area of a rhombus can be evaluated in several ways.

### Area of Rhombus Formula

First Method: The area of a rhombus is equal to the half the product its diagonals.

Let ABCD be a rhombus and its diagonals be d_{1} and d_{2}. If the area is A,

A = 12 d_{1}d_{2} square unit

If the side of a rhombus ABCD is a and its two diagonals are d_{1} and d_{2} respectively and its area is A,

A = 12 d_{1}d_{2} square unit

Second Method: It is known, the area of a parallelogram is equal to the product of its base and height. A rhombus is always a parallelogram.

So, the area of rhombus is found by product its base and height.

Let the side of a rhombus ABCD be a and its base and height be b and h respectively. If the area is A,

A = a.h square unit

If the length of side of rhombus ABCD is a and its height and area are h and A respectively,

A = ah square unit

Third Method: Area of a rhombus can be found by multiplying sine of any angle to the square drawn on its length of the side.

Let ABCD be a rhombus and its length of the side be a. If the area is A,

A = a^{2}sinA square unit

If the angle is θ A = a^{2}sinθ square unit

Fourth Method: Let a and BD be the side length and a diagonal of a rhombus ABCD respectively.

□ABCD = △ABD + △BCD

or, □ABCD = △ABD + △ABD [∵ △ABD ≅ △BCD]

or, □ABCD = 2△ABD

or, □ABCD = 2. 12 a. a sinA

∴ □ABCD = a^{2}sinA

If a is the side and ∠A is the adjacent angle of a rhombus ABCD,

Area of rhombus = a^{2}sinA square unit

## Properties of Rhombus

- The opposite angles of rhombus are equal to each other.
- Two diagonals are perpendicular to each other.
- All sides are equal in length.
- Two diagonals bisect the opposite angles of a rhombus.
- The two diagonals bisect each other.
- All its interior angles add up to 360°.
- Its area is half the product of two diagonals.
- The perpendicular distance between the two opposite sides is considered as the height of a rhombus.
- The two diagonals divides the rhombus into four congruent triangles.