Perimeter of Rhombus

The perimeter of a rhombus is the total distance covered around the edge of the rhombus on a two-dimensional plane. Since the perimeter measures length or distance, its unit is always linear, such as cm, m, km, ft.

The different properties of a rhombus allow us to find its perimeter in different ways. This article will precisely deal with the definition and the ways to find the perimeter of a rhombus with examples.

Formulas

The different properties of a rhombus allow us to find its perimeter in different ways. The 3 ways to find the perimeter of a rhombus are given below.

With Side Length

This is the most common method used to find the perimeter of a rhombus. The formula to calculate the perimeter of a rhombus given the length of a side is as follows:

Let us solve an example to understand the concept better.

Now, there are other ways to calculate the perimeter of a rhombus when side length is not known.

With Diagonals

The formula to calculate the perimeter of a rhombus given the length of the diagonals is as follows:

Derivation

Let us consider  ΔAOB in ▱ABCD in the above figure

AO = d₁/2 and OB = d₂/2

Let the side length = ‘s’

Applying the Pythagorean theorem for ΔAOD, we get

s² = (d₁/2)² + (d₂/2)²

s²= d₁²/4+ d₂²/4

s = √( d₁² + d₂²)/2

As we know, perimeter of a rhombus (P) = 4s

∴ P = 4 × √( d₁² + d₂²)/2

= 2√( d₁²+ d₂²)

With Diagonal and Vertex Angle

There is no direct formula to find the perimeter of a rhombus when any one diagonal and any vertex angle are known.

Before we solve for the perimeter, let us recapitulate 2 simple trigonometric rules for a right triangle:

sinθ = Opposite/Hypotenuse, cosθ = Adjacent/Hypotenuse

Let us now solve a problem to find the perimeter of a rhombus using diagonal and vertex angle

What is the perimeter of rhombus ABCD as shown in the figure below?

Here, diagonal = 6 cm.

Let’s draw another diagonal BD

Half of the diagonal AC is AO = 3 cm

∠ ABD = θ = 30° (as diagonals are angle bisectors)

Let ‘s’ be the side length. In ΔAOB,

sinθ = 3/s, here s = hypotenuse, opposite side  = 3 cm

s = 3/sinθ

= 3/sin(30)

= 3 ÷ 1/2

= 6 cm

As we know,

Perimeter of a rhombus (P) = 4s = 4 × 6 = 24 cm

This is how we derive the perimeter of a rhombus when the diagonal and vertex angle is known.

Now, let us solve an example.

Let’s draw another diagonal WY