Rhombus Formula

Mensuration is a branch of geometry that studies or measures the area, perimeter, and volume of two-dimensional or three-dimensional objects and constructions. Mensuration comprises fundamental mathematical formulae and, in certain circumstances, algebraic expressions.

Rhombus 

A rhombus is a diamond-shaped quadrilateral with equal sides but unequal angles of inclination between these two sides. It has four sides that are all the same length since it is a quadrilateral. 

Rhombus

Properties of a rhombus

• All of the sides are the same length, and opposite sides are parallel to one another.
• Adjacent angles add up to 180°, but opposed angles remain constant.
• The diagonals are perpendicular to each other and bisect the angles between the sides, i.e. the vertex angles.
• The total of the angles in the Rhombus is 360°.
• If each vertex angle is 90°, the rhombus is a square.

Formulae of Rhombus

The formulae of the rhombus include the formula for the area in different ways consisting of different formulas, the formula also includes the perimeter of the rhombus. Let’s take a look at these formulae,

Area of Rhombus 

The entire space covered or encompassed by a rhombus on a two-dimensional plane is defined as its area. The area of a rhombus may be computed using three distinct methods: diagonal, base and height, and trigonometry.

• I^st Case By using Diagonal: It is half of the product of the diagonal lengths.

Area of Rhombus  = (d₁ × d₂)/2 sq. units

Where, d₁ is the length of diagonal 1 and d₂  is the length of diagonal 2.

• 2^nd Case By using Base and Height: The base of a rhombus is one of its sides, and the height is the perpendicular distance from the chosen base to the opposing side.

Area of a Rhombus = base × height sq units

Where, b is the length of any side of the rhombus and h is the height of the rhombus . 

• 3^rd case By using Trigonometry 

Area of Rhombus: (side )² × sin(A) sq. units

Here square the side of Rhombus 

And Sin (A) is the interior angle.

Perimeter of Rhombus 

 A rhombus’ perimeter is the sum of its four sides or It is the product of the length of one side by 4.

Hence the perimeter of the rhombus formula = 4a, where ‘a’ is the side.

Perimeter of Rhombus = side + side + side + side = 4s.

• Rhombus Perimeter Using Diagonal Lengths  

Given a horizontal diagonal length of a and a vertical diagonal length of b, the perimeter is calculated as follows:

P = √(a² + b²) × 2

Sample Questions 

Question 1: What is the area of the rhombus for which the length of diagonals is  6 cm, 8 cm.

Solution:

Given lengths of diagonals,

Diagonal (D₁) = 6cm

Diagonal (D₂) = 8cm

By using Diagonal formula : Area of rhombus = (d₁ × d₂)/2 sq. units

= (1/2) × 6 × 8

= (1/2) × 48

= 24

So the area of rhombus is 24 cm².

Question 2: Calculate the area of a rhombus (using base and height) if its base is 6 cm and height is 2 cm.

Solution: 

Given,

Base (b) = 6 cm

height of rhombus(h) = 2 cm

Now,

Area of the rhombus(A) = base × height 

= 6 × 2

= 12 cm²

Question 3: Find the diagonal of a rhombus if its area is 120 cm² and the length measure of the longest diagonal is 12 cm.

Solution:

Given: Area of rhombus = 120 cm² and Diagonal  d₁ = 12 cm.

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

120 = (12 × d₂)/2

120 = 6 × d₂

Or d₂ = 120/6

d₂ = 20

Therefore, the Length of another diagonal is 20 cm.

Question 4:  Find the perimeter of a rhombus whose side is 7 cm.

Solution: 

Given side s = 7 cm

Therefore, Perimeter of Rhombus: 4 × s

So, Perimeter (P) = 4 × 7 cm = 28 cm

Question 5: Find the side length of a rhombus whose perimeter is given as 60cm.

Solution: 

Given Perimeter(P) = 60 cm

Perimeter = 4 × side

Side = P/4

So, side = 60/4 

= 15 cm

Hence, the length of rhombus is 15 cm.

Question 6:  Find the perimeter of the rhombus given the diagonal lengths are 3 cm and 4 cm respectively.

Solution:

When diagonal lengths are Given a = 3 cm, b = 4 cm  

Perimeter(P) = 2 × √(a² + b²) 

= 2 × √(3² + 4²)

= 2 × √(9 + 16)

= 2 × 5

= 10 cm

Question 7: Find the perimeter of a rhombus whose side is 3.5 cm.

Solution: 

Given that side s = 3.5 cm

Perimeter of Rhombus is given by: 4 × s

So, Perimeter (P) = 4 × (3.5) cm 

= 14 cm