Perimeter of Rhombus
The perimeter of a rhombus is the sum of all its sides. The unit of the perimeter is same as that of a side. Let us recall what is a rhombus. It is a quadrilateral in which all four sides are of the same measure. A rhombus is always a parallelogram (but a parallelogram doesn't need to be a rhombus) as its opposite sides are parallel. All the angles of a rhombus don't need to be right angles. Hence, a rhombus doesn't need to be a square though all its four sides are equal. Thus, a rhombus doesn't need to be a square but a square is always a rhombus. The properties of a rhombus are:
 All sides are equal.
 Opposite sides are parallel.
 Opposite angles are equal.
 The sum of any two adjacent angles is 180^{o}.
 Diagonals bisect each other at right angles.
 Each diagonal bisects the vertex angles.
What Is the Perimeter of a Rhombus?
The perimeter of a rhombus is the length of the outline of the rhombus and is thus obtained by adding the lengths of all its sides. Here, we derive the formulas to find the perimeter of a rhombus in the following cases:
 Perimeter of a rhombus with sides
 Perimeter of a rhombus with diagonals
 Perimeter of a rhombus with one angle and one diagonal
Here are the formulas to find the perimeter of a rhombus in different cases.
But how to derive them? Let us see.
Perimeter of Rhombus Formula With Sides
As discussed earlier, the perimeter of a rhombus is the sum of the lengths of all its sides. We know that the sides of a rhombus are of equal lengths. Let us consider a rhombus of side length 'a' units. Then the perimeter of the rhombus is a + a + a + a which is 4a. Thus, the perimeter of the rhombus formula is,
Perimeter of rhombus = 4a units.
Example: Find the perimeter of a rhombus of side length 10 units.
Solution:
The side length of the given rhombus, a = 10 units.
Its perimeter = 4a = 4(10) = 40 units.
Perimeter of Rhombus Formula With Two Diagonals
Sometimes, we are not given the side length of the rhombus, instead, we are given the lengths of the diagonals. In this case, we find the side length of the rhombus using the Pythagoras theorem. Here, we will make use of the following properties:
 A rhombus is divided into 4 congruent rightangled triangles by its two diagonals.
 The diagonals bisect each other at right angles.
Let us consider a rhombus ABCD with diagonals \(d_1\) and \(d_2\) and with side length 'a'.
Since the diagonals bisect at right angles, by applying the Pythagoras theorem for triangle AOD, we get
\(a^2 = \dfrac{d_1^2}{4}+\dfrac{d_2^2}{4}\)
\(a =\dfrac{ \sqrt{d_1^2+d_2^2}}{2}\)
From the previous section we know that
Perimeter of the rhombus = 4a
Substituting \(a =\dfrac{ \sqrt{d_1^2+d_2^2}}{2}\) in the above formula,
Perimeter of rhombus = 4 \(\dfrac{ \sqrt{d_1^2+d_2^2}}{2}\) (or)
Perimeter of rhombus = \(2\sqrt{d_1^2+d_2^2}\)
Note: We do not need to remember this formula. Instead, we can use this procedure of using the Pythagoras theorem to find the side length of the rhombus using the diagonals and then we can apply the perimeter of rhombus formula to be 4 × side length.
Perimeter of Rhombus Using One Diagonal and One Angle
Here are the steps to find the perimeter of a rhombus when one diagonal and one vertex angle are given.
 Divide the rhombus into 4 congruent triangles using the diagonals and consider one triangle among them.
 Since the diagonals interest at right angles, mark the angle at the point of intersection of the diagonals to be a right angle.
 Since a diagonal bisects the angles at the vertices, half of the given angle will become one of the vertex angles of the triangle that we have considered.
(Here, you may need to consider the fact that the opposite angles of a rhombus are equal).  Since the diagonals bisect each other, half of the given diagonal will become one of the sides of the triangle that we have considered.
 Find the side length, 'a' using one of the trigonometric ratios.
 Find the perimeter of the rhombus using the formula,
Perimeter of the rhombus = 4a
We will see the applications of all the formulas of the perimeter of the rhombus in the following section.
Solved Examples on Perimeter of Rhombus

Example 1: Find the perimeter of the rhombus with diagonals 8 inches and 6 inches.
Solution:
Method 1:
The lengths of the diagonals of the given rhombus are, \(d_1\) = 8 in and \(d_2\) = 6 in.
Using the perimeter of rhombus formula using diagonals,
Perimeter = 2 \(\sqrt{d_1^2+d_2^2}\)
Perimeter = 2 \(\sqrt{8^2+6^2}\) = 2 \(\sqrt{100}\) = 2 (10) = 20 in.
Method 2:
Let us assume that the side length of the rhombus be 'a' inches.
Since the diagonals bisect each other at right angles,
By Pythagoras theorem,
a^{2} = 3^{2} + 4^{2} = 25
a = 5 in.
Thus, the perimeter of the rhombus is, 4a = 4 (5) = 20 in.
Answer: The perimeter of the given rhombus = 20 in.

Example 2: ABCD is a rhombus with ∠ABC = 60^{o }and AC = 20 units. Find the perimeter of ABCD.
Solution:
Let us assume that the side length of the rhombus ABCD is 'a' units.
Here we will use the procedure of "perimeter of a rhombus with one angle and one diagonal" to find the perimeter.
We know that a diagonal bisects the vertex angles and the diagonals bisect at right angles. Thus,
By applying Sine to the triangle AOB,
sin 30^{o} = 10/a
Using the trigonometric table,
1/2 = 10/a
Cross multiplying,
a = 20
Thus, the perimeter of the rhombus is, 4a = 4 (20) = 80 units.
Answer: The perimeter of the rhombus = 80 units.
FAQs on Perimeter of Rhombus
What Is the Perimeter of a Rhombus?
The perimeter of a rhombus is the sum of all its sides. If 'a' represents the side length of a rhombus, then its perimeter (as its all 4 sides are equal) is a + a + a + a = 4a units.
What Is the Perimeter of a Rhombus Formula When Side Length Is Given?
Since all 4 sides of a rhombus are equal, its perimeter is obtained just by multiplying its side by 4. i.e., the perimeter of a rhombus formula for a rhombus of side 'a' units is 4a units.
How To Find Perimeter of Rhombus When Diagonals Are Given?
If \(d_1\) and \(d_2\) are the diagonals of a rhombus, then the formula to find its perimeter is 2 \(\sqrt{d_1^2+d_2^2}\). Instead, we can also divide the rhombus into 4 congruent rightangled triangles, apply the Pythagoras theorem to one of the triangles to find the side length 'a' of the rhombus, and find the perimeter using the formula 4a.
How To Find the Perimeter of Rhombus?
There are multiples formulas for finding the perimeter of a rhombus.
 The perimeter of a rhombus of side length 'a' is 4a.
 The perimeter of a rhombus of diagonals \(d_1\) and \(d_2\) is 2 \(\sqrt{d_1^2+d_2^2}\).
How To Find the Perimeter of a Rhombus When Angle and Diagonal Are Given?
Steps to find the perimeter of a rhombus when one diagonal and one angle are given:
 Divide the rhombus into 4 congruent triangles using the diagonals and consider one triangle among them.
 Since the diagonals interest at right angles, mark the angle at the point of intersection of the diagonals to be a right angle.
 Since a diagonal bisects the vertex angles, half of the given angle will become one of the vertex angles of the triangle that we have considered.
(Here, you may need to consider the fact that the opposite angles of a rhombus are equal).  Since the diagonals bisect each other, half of the given diagonal will become one of the sides of the triangle that we have considered.
 Find the side length, 'a' by applying any trigonometric ratio to the triangle that we have considered.
 Find the perimeter of the rhombus using the formula 4a.
How To Find the Side Length When the Perimeter of a Rhombus Is Given?
We know that the perimeter of a rhombus of side length 'a' is 4a. Thus, the side length of the rhombus can be obtained by dividing its perimeter by 4.
How To Find a Diagonal of a Rhombus When Its Perimeter and the Other Diagonal Are Given?
We know that if \(d_1\) and \(d_2\) are the diagonals of a rhombus, then the formula to find its perimeter is, P = 2 \(\sqrt{d_1^2+d_2^2}\). We can use this equation to solve for a diagonal (say \(d_1\)) when the perimeter (P) and the other diagonal (say \(d_2\)) are given.