## Formulas of Rhombus

Wondering how to find the perimeter of a rhombus? Just like any polygon, the perimeter of the rhombus is the total distance around the outside, which can be simply calculated by adding up the length of each side. In the case of a rhombus, all four sides are of similar length, thus the perimeter is four times the length of a side. Or as a formula: Perimeter of rhombus = 4 a = 4 × side. Here, ‘a’ represents each side of a rhombus.

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### Perimeter of Rhombus Formula

A rhombus is a 2-dimensional (2D) geometrical figure which consists of four equal sides. Rhombus has all sides equal and its opposite angles are equivalent in measurement. Let us now talk about the rhombus formula i.e. area and perimeter of the rhombus.

### The Perimeter of a Rhombus

The perimeter is the sum of the length of all the 4 sides. In rhombus all sides are equal.

Thus, the Perimeter of rhombus = 4 × side

So, P = 4s

In which,

S = length of a side of a rhombus

### Area of Rhombus Formula

The area of a rhombus is the number of square units in the interior of the polygon. The area of a rhombus can be identified in 2 ways:

i) Multiplying the base and height as rhombus is a unique kind of a parallelogram.

Area of rhombus = b × h

In which,

B = base of the rhombus

H = height of the rhombus

ii) By determining the product of the diagonal and dividing the product by 2.

Area of rhombus formula = 1/2 * d₁ * d₂

In which,

d_{1} × d_{2} = diagonal of the rhombus

### Derivation of Area of Rhombus

Let MNOP is a rhombus whose base MN = b, PN ⊥ MO, PN is a diagonal of rhombus = d₁, MO is diagonal of rhombus = d₂, and the altitude from O on MN is OZ, i.e., h.

Area of Rhombus MNOP = 2 Area of ∆ MNO

= 2 × ½ MN × OP sq units.

= 2 × ½ b × h sq. units

= base × height sq. units

Area of rhombus = 4 × Area of ∆ MZP

= 4 × ½ × MZ × ZN sq. units

= 4 × ½ × d_{2} × ½
d1 sq. units

Thus,

= 4 × ⅛ d_{1} × d_{2} sq. units

= ½ × d_{1} × d_{2}

Hence, the area of a rhombus

= ½ (product of diagonals) sq. units.

### Fun Facts

A rhombus consists of an inscribed circle

In a rhombus, all sides are equal, just as a rectangle has all angles equal.

A rhombus has opposite angles equivalent to each other, while a rectangle has opposite sides equal.

### Solved Examples

Example:

Evaluate the area of the rhombus MNOP having each side equal to 15 cm and one of its diagonals equal to 18 cm.

Solution:

Given:-

MNOP is a rhombus in which MN = NO = OP = PM = 15 cm

MO = 18 cm

Thus, MZ = 9 cm

In ∆ MZP,

MP² = MZ² + ZP²

⇒ 15² = 9² + ZP²

⇒ 225 = 81 + ZP²

⇒ 144 = ZP²

⇒ ZP = 12

Hence, NP = 2 P

= 2 × 12

= 24 cm

Now, to find out the area of rhombus, we will apply the formula i.e.

= ½ × d₁ × d₂

= ½ × 18 × 24

= 216 cm²

Example:

Find the perimeter of a rhombus MNOP whose diagonals measure 20 cm and 24 cm respectively?

Solution:

Given:-

d1 = 20 cm

d2 = 24 cm

MZ= 20/2 = 10cm

NZ= 24/2= 12 cm

∠MZP = 90°

Now applying the Pythagorean Theorem, we know that

MN2 = MZ2 + NZ2

MN = √(100 + 144)

= 15.62 cm

Since, MN = NO = OP = MP,

Therefore, Perimeter of MNOP = 15.62 × 4 = 62.48 cm.

Q1. What is a Rhombus?

Answer: A rhombus is a special kind of a parallelogram with its entire sides equal. A square is a kind of rhombus with all of the angles being equal and also all of the sides. Moreover, both rhombuses and squares have perpendicular diagonal bisectors that divide each diagonal into two equal segments, and also dividing the quadrilateral into four equal right triangles.

Having said that, we know the Pythagorean Theorem would work sufficiently in this situation, using half of each diagonal as the two legs of the right triangle.

Q2. What are the Properties of a Rhombus?

Answer: Following are the properties of a rhombus:-

Rhombus has all sides equal.

Opposite angles of a rhombus are equal.

Sum of adjacent angles are supplementary i.e. (∠N + ∠O = 180°).

If one angle of a rhombus is right, then all angles are right.

Each diagonal splits it into two congruent triangles.

In a rhombus, diagonals intersect each other and are perpendicular to each other.

Q3. What are the Dual Properties of a Rhombus?

Answer: Note that the dual polygon of a rhombus is a rectangle:

A rhombus consists of an axis of symmetry across each pair of opposite vertex angles, whereas a rectangle has an axis of symmetry across each pair of opposite sides.

The diagonals of a rhombus bisect at equal angles. On the other hand, the diagonals of a rectangle are equivalent in length.

The geometrical shape formed by connecting the midpoints of the sides of a rhombus is a rectangle, and reciprocally.