Perimeter of Rhombus Formula: Derivation & Properties

The perimeter of a Rhombus formula is 4 x s (Here, s = the side length of the rhombus). Rhombus is a flat-shaped two dimensional structure having four equal angles and four equal sides. The four angles are not 90 degrees necessarily. 

This shape is also considered to be a diamond-shaped structure. The perimeter of the rhombus is actually the total distance covered along the border of it. 

Key Terms: Rhombus, Perimeter, Pythagoras Theorem, Adjacent Angles, Bisect, Diagonals

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Perimeter of Rhombus

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According to the definition of the Rhombus, 

Where,

s = the side length of the rhombus.

Hence, P = s + s + s + s = 4s

Rhombus

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Area of the Rhombus

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The area of rhombus can be determined in two ways:

i) By multiplying the height & base of the rhombus

Area = b x h [b = base of rhombus; h = height of rhombus]

ii) By determining the product of the diagonal of the rhombus & dividing it by 2

Area = ½ x d₁ x d₂ [d₁ & d₂ are the diagonals of the rhombus]

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Derivation of the Area of Rhombus

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If MNOP is considered a Rhombus with base MN = b, PN ⊥ MO, MO is a diagonal of rhombus = d₂; PN is a diagonal of rhombus = d₁; altitude from O on MN is OZ which is = h.

• Then, the area of the rhombus MNOP = 2 x area of ∆ MNO

= 2 x ½ MN x OP

= 2 x ½ b x h (sq. Units)

• The area of the rhombus = 4 x Area of ∆ MZP

= 4 x ½ x MZ x ZN 

= 4 x ½ x d₂ x ½ d₁ (sq. Units)

= 4 x 1/8 d₁ x d₂

The area of a rhombus = ½ x d₁ x d₂

Finding Perimeter of Rhombus Using Diagonal Lengths

Given, in a rhombus PQRS, a as the horizontal diagonal length and b as vertical diagonal length and center o.

Let’s take triangle POS, then the length of
OP = a/2

OS = b/2

If we assume the side of the rhombus as = X

Using Pythagoras theorem, 

X² = a²/2 + b²/2

X² = (a² + b²)/4

X = \(\sqrt{ \frac{( a^2+b^2)}{2}}\)

By substituting the value of X in the rhombus formula = 4X, we get, 

4 x \(\sqrt{ \frac{( a^2+b^2)}{2}}\)

So, the perimeter will be = 2 x \(\sqrt{ \frac{( a^2+b^2)}{2}}\)

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Properties of Rhombus

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Rhombus is known to be a square due to the following properties of Rhombus:

• It’s all sides have equal length
• It’s opposite sides are parallel
• Its diagonals bisect each other at 90 degrees.
• Two adjacent angles forms a total of 180 degree
• It’s altitude is the distance at right angles between the two parallel sides

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Things to Remember

• Diagonals of a rhombus divide each other at right angles or 90 degrees.
• The perimeter of a rhombus is said to be the sum of all four sides
• The sum of all the angles of a rhombus is 360 degrees
• The rhombus share the same properties as that of a square. Hence, rhombus is also said to be a square.
• Rhombus differs from rectangle in a way as rhombus has opposites angles as equivalent while rectangle has opposite sides as equivalent