Rhombus Formula

The rhombus formulas are used to find the unknown dimensions associated with a rhombus. A rhombus is a quadrilateral with 4 equal sides. The basic rhombus formulas depend on the parameters that are given, like the side length, the two diagonal lengths, the area, the perimeter, and so on. Let us learn about the basic rhombus formulas along with a few solved examples.

What is the Rhombus Formula?

The basic rhombus formulas include the following:

• Perimeter of Rhombus Formula = 4 × a, where 'a' is the side.
• Area of a Rhombus Formula = 1/2 × d₁ × d₂ where, d₁ and d₂ are the diagonals,
• Rhombus Area Formula with Sides = a² × sin x, where, a is the side length of the rhombus, and x is the interior angle
• Side of Rhombus Formula (a) = [√(p² + q²)] ÷ 2, where 'p' and 'q' are the two diagonals.
• Diagonal of Rhombus Formula, Diagonal₁ = (2 × Area) ÷ Diagonal₂

Let us learn more about these rhombus formulas now. Observe the rhombus shown below in which 'a' is the side and 'd₁' and 'd₂' represent the diagonals of the rhombus.

Perimeter of Rhombus Formula

The perimeter of a rhombus is the total length of the boundary of the rhombus and is the sum of its four sides. Therefore, the perimeter of rhombus formula = 4 × a, where 'a' is the side. Let us understand this with the help of an example.

Example: Find the perimeter of a rhombus whose side length is 6 units.

Solution: Given, side length = 6 units.

Let us find the perimeter using the perimeter of rhombus formula = 4 × a, where 'a' is the side. After substituting the value of a = 6, we get

perimeter of rhombus = 4 × a = 4 × 6 = 24 units

Area of a Rhombus Formula

To find the area of a rhombus, we find the total space enclosed by a rhombus. The area of the rhombus can be calculated by the following rhombus formula:

Area of rhombus formula = 1/2 × d₁ × d₂

where,

• d₁ and d₂ are the diagonals,

Let us understand this with the help of an example.

Example: Find the area of a rhombus in which the length of the diagonals are 6 units and 8 units respectively.

Solution: Given, diagonal₁ = 6 units, diagonal₂ = 8 units,

Let us find the area using the Area of rhombus formula = 1/2 × d₁ × d₂

After substituting the value of d₁ = 6, and d₂ = 8, we get,

Area of rhombus = 1/2 × 6 × 8 = 24 square units.

Rhombus Area Formula with Sides

The area of the rhombus can be calculated by another formula. When the length of the side of the rhombus and interior angle are given, Area of rhombus formula = a² × sin x

where,

• a is the length of the side of the rhombus,
• x is the interior angle

Side of Rhombus Formula

The side of rhombus formula depends on the parameters that are given. For example, if the diagonals are given, then the Side of rhombus formula (a) = [√(p² + q²)] ÷ 2

where 'p' and 'q' are the two diagonals. Let us understand this with the help of an example.

Example: Find the length of the side of a rhombus in which the length of the diagonals are 8 units and 10 units respectively.

Solution: Given, (p) diagonal₁ = 8 units, (q) diagonal₂ = 10 units,

Side of rhombus formula, (a) = √(8² + 10²) ÷ 2

a = √(64 + 100) ÷ 2

a = (√164) ÷ 2

a = 12.8 ÷ 2 = 6.4 units

Therefore, the side of the rhombus is 6.4 units.

Diagonal of Rhombus Formula

The length of 1 diagonal of a rhombus at a time can be calculated if the area and the other diagonal is known.

The diagonal of rhombus formula is, Diagonal₁ = (2 × Area) ÷ Diagonal₂

The same formula is applicable to the second diagonal. This means, Diagonal₂ = (2 × Area) ÷ Diagonal₁

Let us understand all these rhombus formulas using a few solved examples.

Examples on Rhombus Formulas

Example 1: Find the area of a rhombus in which the diagonal lengths are 10 units and 8 units respectively.

Solution:

We need to find the Area of the rhombus
Let us use the area of a rhombus formula, with the two diagonal lengths
Area of the rhombus = 1/2 × d₁ × d₂
Area of the rhombus = 1/2 × 10 × 8
Area of the rhombus = 40

Answer: The area of the rhombus is 40 square units.

Example 2: Using the rhombus formula find the area of the rhombus with an interior angle of 30° and length of the side 5 inches.

Solution:

We need to find the Area of the rhombus. Let us use the area of a rhombus formula with the interior angle and the side length.

Interior angle (a) = 30 degrees, side length = 5 inches

Using the rhombus formula,

Area of the rhombus = a² × sin x

Area of the rhombus = 5² × sin 30

Area of the rhombus = 25 × 1/2

Area of the rhombus = 12.5

Answer: The area of the rhombus is 12.5 unit inches.

Example 3: A rhombus has a side of length 16 units. Can you find the perimeter of the rhombus using the perimeter of rhombus formula?

Solution:

We know the side length of the rhombus = 16 units.
Since all sides of a rhombus are equal, all four sides are equal to 16 units.
Perimeter = 4 × side = 4 × 16 = 64 units

Answer: The perimeter of the rhombus is 64 units.