# 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.

Find the perimeter of a rhombus with a side of 4 cm.

Solution:

As we know,
P = 4s, here s = 4 cm
= 4 Ã— 4
= 16 cm

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  Î”AOB in â–±ABCD in the above figure

AO = d1/2 and OB = d2/2

Let the side length = ‘s’

Applying the Pythagorean theorem for Î”AOD, we get

s2 = (d1/2)2 + (d2/2)2

s2= d12/4+ d22/4

s = âˆš( d12 + d22)/2

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

âˆ´ P = 4 Ã— âˆš( d12 + d22)/2

= 2âˆš( d12+ d22)

Finding the perimeter of a rhombus when DIAGONALS are known

Find the perimeter of a rhombus with diagonals 8 cm and 12 cm

Solution:

As we know,
P = 2âˆš( d12+ d22), here d1 = 8 cm, d2 = 12 cm
= 2 Ã— âˆš(64 + 144)
= 28.844 cm

### 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:

sinÎ¸ = Opposite/Hypotenuse, cosÎ¸ = 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

âˆ  ABD = Î¸ = 30Â° (as diagonals are angle bisectors)

Let ‘s’ be the side length. In Î”AOB,

sinÎ¸ = 3/s, here s = hypotenuse, opposite side  = 3 cm

s = 3/sinÎ¸

= 3/sin(30)

= 3 Ã· 1/2

= 6 cm

As we know,

Perimeter of a rhombus (P) = 4s = 4 Ã— 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.

Finding the perimeter of a rhombus when ONE DIAGONAL and VERTEX ANGLE are known

What is the perimeter of rhombus WXYZ having the longer diagonal as 6 cm and the vertex angle as 70Â°

Solution:

Half of diagonal XZ is XO = 3 cm
âˆ  WXZ = Î¸ = 35Â° (as diagonals are angle bisectors)
In Î”WXO, WX is the hypotenuse
Now, cosÎ¸ = 3/s, here s = hypotenuse, adjacent sideÂ  = 3 cm
âˆ´ s = 3/cos(35)
= 3/0.819
= 3.663 cm
As we know,
P = 4s, here s = 3.663 cm
= 4 Ã— 3.663
= 14.652 cm

Let’s draw another diagonal WY