Area of Rhombus

A rhombus is a quadrilateral figure where all four sides are of equal length. The opposite angles are equal and its diagonals bisect each other at 90 degrees. Due to this, the rhombus can be divided into four right triangles. The area of the rhombus is the total space it encloses.

To find the area of the rhombus, we use the formula:- (d₁ x d₂) / 2, where d₁ and d₂ are the lengths of its diagonals. Now let’s see how the formula is derived.

Derivation of the formula:-

• The diagonals d₁ and d₂ intersect at right angles, dividing the rhombus into four right angle triangles.

• Let one triangle be △AOB. 

• The area of the △AOB = 1 / 2 × b × h = 1 / 2 x AO x OB

• The diagonals d₁ and d₂ intersects at the point O. So, AO = d₁/2 and OB = d₂/2.

• When the value of AO and OB is substituted in the formula △AOB =  12 x AO x OB, we get 1 / 2 × d₁/2 × d₂/2.

• Since rhombus has four sides, the total area will be 4 x Area of  △AOB

• Multiplying 4 with the value of area △AOB, we get the area of rhombus.

• Area of rhombus = 4 x Area of △AOB = 4 × 1 / 2 × d₁/2 × d₂/2 = 1 / 2 × d₁ × d₂

Therefore, the area of the rhombus = 12 × d1 × d2

We can find the area of the rhombus using three methods, where the formula using diagonals is commonly used. The area of the rhombus is found using three methods. They are:

• Method using the diagonals
   
• Method using the base and height
   
• Method using the side and angle

Now let’s discuss the methods mentioned.

If the diagonals d1 and d2 are given, we find the area of the rhombus using the formula Area  = 1 / 2 x d₁ x d₂

For example, d₁ and d₂ are 4 cm and 20 cm. What will be the area of the rhombus?

Area  = 1 / 2 × d₁ × d₂  

= 1 / 2 × 4 × 20 

= 2 × 20 

= 40

The area of the rhombus is 40 cm2

When the base and perpendicular height is given, we find the area of rhombus using the formula Area = base × height

For example, the base and height is given as 34 cm and 12 cm. What will be the area of the rhombus?

Area = base × height = 34 × 12 = 408

The area of the rhombus is 408 cm².

If the length of one side is ‘a’ and the interior angle is ‘θ’, the area of the rhombus = a² x sin θ.

For example, if ‘a’ is 6 and θ is 30 degrees, what will be the area of the rhombus?

Area = a² × sin θ = 6² × sin 30 = 36 × 0.5 = 18

Therefore, the area of the rhombus is 18 cm².

We measure the area of a rhombus in square units. The measurement depends on the system used:

In the metric system, the area is measured in square meters (m²), square centimeters (cm²) and square millimeters (mm²)

In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²)

Since a rhombus is a quadrilateral having equal sides, its diagonals that bisect each other are also perpendicular, which results in three principal methods by which the area can be calculated. Take a look at the special cases:

Case 1: Use of Diagonals

If diagonals are given to find the area of the rhombus, use the formula
Area = 1 / 2 × d₁ × d₂, where d₁ and d₂ are the diagonals.

Case 2: Use of base and height

If there’s base and height given to find the area of the rhombus, we use the formula Area = base × height

Case 3: Using Trigonometry

If the length of a side ‘s’ and an angle is given, we find the area of the rhombus using the formula Area = s² × sin θ

To make sure that you get correct results while calculating the area of the rhombus, there are some tips and tricks you should know about. Here are some:-

• The height of the rhombus is different from the side length.

• The area can be calculated in three ways, depending on the information provided. There are formulas to find the area when the diagonals, base and height, side length, and angle are given.

• Use the Pythagoras theorem to find the second diagonal if the side length and the other diagonal are known.

• Area: The total space occupied by a shape. It tells us how much surface a shape covers.

• Quadrilateral: A geometrical shape with four sides. For example, shapes like squares, rhombuses, and parallelograms are quadrilaterals.

• Imperial System: Measurement system which is commonly used in the United States of America. The area is measured in square inches (in²), square feet (ft²), and square yards (yd²).