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Last modified on August 3rd, 2023

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.

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.

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.

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

Let us consider Î”AOB in â–±ABCD in the above figure

AO = d_{1}/2 and OB = d_{2}/2

Let the side length = ‘s’

Applying the Pythagorean theorem for Î”AOD, we get

s^{2} = (d_{1}/2)^{2} + (d_{2}/2)^{2}

s^{2}= d_{1}^{2}/4+ d_{2}^{2}/4

s = âˆš( d_{1}^{2} + d_{2}^{2})/2

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

âˆ´ P = 4 Ã— âˆš( d_{1}^{2} + d_{2}^{2})/2

= **2âˆš(** **d _{1}^{2}+**

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( d_{1}^{2}+ d_{2}^{2}), here d_{1} = 8 cm, d_{2} = 12 cm

= 2 Ã— âˆš(64 + 144)

= 28.844 cm

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

Last modified on August 3rd, 2023