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
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
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
d1 × d2 = 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.
= 2 × ½ MN × OP sq units.
= 2 × ½ b × h sq. units
= base × height sq. units
= 4 × ½ × MZ × ZN sq. units
= 4 × ½ × d2 × ½
d1 sq. units
= 4 × ⅛ d1 × d2 sq. units
= ½ × d1 × d2
Hence, the area of a rhombus
= ½ (product of diagonals) sq. units.
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.
Evaluate the area of the rhombus MNOP having each side equal to 15 cm and one of its diagonals equal to 18 cm.
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
Find the perimeter of a rhombus MNOP whose diagonals measure 20 cm and 24 cm respectively?
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.