Perimeter of Rhombus Formula
In mensuration, the perimeter of a is defined as the sum of lengths of all the sides of the quadrilateral around the border. So perimeter of the rhombus is defined as the sum of all 4 sides of the rhombus.
Rhombus is a diamond-shaped quadrilateral whose all sides are equal but each angle inclined between these two sides is not equal. Since it is a quadrilateral it has four sides and all four sides are of equal length. It has the following properties.
- All the sides are equal in length and opposite sides are parallel to each other.
- Adjacent angles sum to 180 degrees and opposite angles remain the same.
- The diagonals bisect each other perpendicularly and bisect the angles between the sides i.e vertex angles.
- The Sum of all the angles in the Rhombus is 360 degrees.
- The rhombus is a square if each vertex angle is equal to 90 degrees.
The shape of the Rhombus:
Perimeter of Rhombus using Side Lengths
Formula according to the definition:
Perimeter of Rhombus = 4×s
where
s is the side length of the rhombus.
Derivation:
Perimeter(P) = s + s + s + s = 4*s
Perimeter of Rhombus using Diagonal Lengths
Given horizontal diagonal length as a and vertical diagonal length as b then perimeter is given by:
P = 2 * √(a2 + b2)
Derivation:
Since diagonals bisect each other at right angles each quadrant forms a right angled triangle and the lengths of the sides i.e base and height are a/2 and b/2 and side of Rhombus as s.
By applying Pythagoras Theorem:
a2/4 + b2/4 = s2 (side)
s = (√(a2 + b2))/2
P = 4 * s = 2 * √(a2 + b2)
Sample Problems
Question 1: Find the perimeter of a rhombus whose side is 8 cm.
Solution:
Given that side s = 8 cm
Perimeter of Rhombus is given by : 4*s
So, Perimeter (P) = 4 * 8 cm = 32 cm
Question 2: Find the side length of a rhombus whose perimeter is given as 36cm.
Solution:
Given Perimeter(P) = 36 cm
P = 4 * s
=> s = P/4
So, s = 36/4 = 9cm
Question 3: Find the perimeter of the rhombus given the diagonal lengths are 6 cm and 8 cm respectively.
Solution:
When diagonal lengths are given :
Given a = 6 cm, b = 8cm
Perimeter(P) = 2* √(a2 + b2) = 2* √(36 + 64) = 2 * 10 = 20 cm
Question 4: Find the length of horizontal diagonal given the side length as 13cm and vertical diagonal length as 24 cm.
Solution:
Since the diagonal bisect at right angles:
Given b = 24 cm and s = 13 cm, a = ?
side(s) is given as
s = (√(a2 + b2))/2
2 * s = (√(a2 + b2))
26 = (√(a2 + 576))
On squaring both sides, 676 = a2 + 576
=> a2 = 100
=> a= 10cm
Question 5: Find the area of the rhombus whose diagonal are of lengths 24cm and 10 cm.
Solution:
Given a = 24 and b = 10cm
Area of the Rhombus is given by A = 1/2 * a * b
= 1/2 * 24 * 10
= 60 cm2
Question 6: Find the perimeter of a rhombus whose side is 2.5 cm.
Solution:
Given that side s = 8 cm
Perimeter of Rhombus is given by: 4*s
So, Perimeter (P) = 4 * (2.5) cm = 10 cm
Question 7: Find the side length of a rhombus whose perimeter is given as 48cm.
Solution:
Given Perimeter(P) = 48 cm
P = 4 * s
=> s = P/4
So, s = 48/4 = 12cm
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