Perimeter of Rhombus – Definition, Formula & Examples

A perimeter is a path that surrounds a 2D shape. The word comes from the Greek words peri and meter. The term is used either for the path or the length of the outline of a shape. Calculating the perimeter has considerable practical applications. The perimeter can be used to calculate the length of fence required to surround a yard or garden.  The perimeter of a rhombus is the sum of all its sides. The rhombus is a quadrilateral in which all four sides are of the same measure.

Let’s learn about the perimeter of rhombus, its formula, and its properties.

Rhombus – A 2D Plane Figure

A rhombus is a special type of parallelogram. In a rhombus, opposite sides are parallel and the opposite angles are equal. Moreover, all the sides of a rhombus are equal in length, and the diagonals bisect each other at right angles. The rhombus is also called a diamond or rhombus diamond.

These properties of the rhombus help in distinguishing the rhombus from other quadrilaterals.

• All four sides are equal
• The opposite sides are parallel
• The opposite angles are equal
• The sum of any two adjacent angles is $180^{\circ}$
• The 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 total length of its boundary and it is calculated by adding the length of all its sides (edges). Since all the sides of a rhombus are equal, the perimeter of a rhombus can also be expressed as ‘four times the sides’.

Perimeter of Rhombus Formula Using Sides

As discussed above, the perimeter of a rhombus is the sum of the lengths of all its sides. Since the sides of a rhombus are of equal lengths if we consider side length as $a$, then the perimeter of the rhombus is $a + a + a + a = 4a$. Thus, the perimeter of the rhombus formula is $P = 4a$.

Examples

Ex 1: Find the perimeter of a rhombus whose side is $2.8 in$.

Length of side of rhombus $a = 2.8 in$.

Perimeter of rhombus $P = 4a = 4 \times 2.8 = 11.2 in$.

Therefore, the perimeter of a rhombus of side $2.8 in$ is $11.2 in$.

Ex 2: Find the length of each side of a rhombus whose perimeter is $67 cm$.

Perimeter of a rhombus $P = 67 cm$

Let the length of each side of the rhombus be $a$.

Therefore, $4a = 67 => a = \frac {67}{4} = 16.75 cm$.

Perimeter of Rhombus Formula Using Diagonals

Consider a rhombus of side length $a$ and the length of the two diagonals as $p$ and $q$. As discussed above the diagonals of a rhombus bisect each other at right angles. So, we get four congruent right triangles with the centre of the rhombus as the common vertex.

Now by Pythagoras theorem, we have $a^{2} = \left(\frac {p}{2}\right)^{2} +  \left(\frac {q}{2}\right)^{2}$

$=> a^{2} = \frac {p^{2}}{4} + \frac {q^{2}}{4} = \frac {p^{2} + q^{2}}{4}$

$=>a = \frac {\sqrt{p^{2} + q^{2}}}{2}$

Therefore, the perimeter of a rhombus becomes $4a = 4 \times \frac {\sqrt{p^{2} + q^{2}}}{2} = 2 \sqrt{p^{2} + q^{2}}$.

Examples

Ex 1: Find the perimeter of a rhombus whose diagonals are $6 cm$ and $8 cm$.

The lengths of the diagonals are $p = 6 cm$ and $q = 8 cm$

Perimeter of rhombus = $2 \sqrt{p^{2} + q^{2}} = 2 \times \sqrt{6^{2} + 8^{2}} = 2 \times \sqrt{36 +64} = 2 \times \sqrt{100}  = 2 \times 10 = 20 cm$

Ex 2: The perimeter of a rhombus is $40 in$. What is the length of the other diagonal, if the length of one diagonal is $12 in$?

The perimeter of a rhombus = $40 in$

One of the diagonal $p = 12 in$

$\text{Perimeter} = 2 \sqrt{p^{2} + q^{2}} => 40 = 2 \times \sqrt{12^{2} + q^{2}}$

$=> \frac{40}{2} = \sqrt{144 + q^{2}} => 20 = \sqrt{144 + q^{2}}$

Squaring both sides

$20^{2} = 144 + q^{2} => 400 = 144 + q^{2} => q^{2} = 400 – 144 => q^{2} = 256$

Taking the square root of both sides

$q = \sqrt{256} = 16$

Therefore, the length of the other diagonal is $16 in$.

Note: You can use the factorization method to find  $\sqrt{256}$. $256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{8}$. Therefore, $\sqrt{256} = \sqrt {2^{8}} = \sqrt {2^4} = 16$.

Conclusion

A rhombus is a quadrilateral with all sides equal and angles not $90^{\circ}$. The perimeter of the rhombus is obtained by using the formula $4a$, where $a$ is the length of each side of the rhombus.

Practice Problems

1. Find the perimeter of a rhombus whose sides are of length
   • $7.8 m$
   • $45 mm$
2. Find the length of each side of a rhombus, if the perimeter is
   • $98 in$
   • $42 cm$
3. Find the perimeter of a rhombus whose diagonals are of lengths
   • $14 mm$ and $48 mm$
   • $40 m$ and $42 m$

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