Perimeter of Parallelogram
The perimeter of a parallelogram is the length of the continuous line formed by its boundary. Its unit is same as that of its sides. A quadrilateral is a closed shape made up of four line segments. A quadrilateral is called a parallelogram if its opposite sides are parallel and are of equal length. Some examples of a parallelogram are rhombus, rectangle, and square. Here are some properties of a parallelogram.
 Opposite sides are equal.
 Opposite angles are equal.
 Diagonals bisect each other.
 Every two adjacent angles are supplementary.
We will learn how to find the perimeter of a parallelogram using some formulas along with a few solved examples here.
What Is the Perimeter of Parallelogram?
The perimeter of a parallelogram is the length of its outline and hence it is equal to the sum of all its sides. But we always might not know all the sides of a parallelogram. Instead, we may be given other information about the parallelogram and are asked to find its perimeter. The perimeter of a parallelogram can be found in the following cases.
 When two adjacent sides are known.
 When one side and diagonals are known.
 When base, height, and any angle are known.
Here are the formulas to find the perimeter of a parallelogram in each of these cases.
But how to derive all these formulas? Let us see.
Perimeter of Parallelogram Formula With Sides
As discussed earlier, the perimeter of a parallelogram is the sum of the lengths of all its sides. We know that the opposite sides of a parallelogram are equal. Let us consider a parallelogram with two adjacent sides to be 'a' and 'b' (then the other two adjacent sides also will be 'a' and 'b' only).
Then the perimeter of the parallelogram is a + b + a + b (or) 2a + 2b (or) 2 (a + b). Thus, the perimeter (P) of a parallelogram with sides is,
P = 2 (a + b) units.
Example: Find the perimeter of a parallelogram whose adjacent sides are 5 units and 9 units.
Solution:
The adjacent sides of the given parallelogram are, a = 5 units and b = 9 units. Then its perimeter (P) is,
P = 2 (a + b)
P = 2 (5 + 9) = 2 (14) = 28 units.
Perimeter of Parallelogram Formula With One Side and Diagonals
Let us consider a parallelogram with sides 'a' and 'b' and diagonals 'x' and 'y'. Assume that the values of the side 'a', and the diagonals 'x' and 'y' are given but the value of 'b' is not given and we are asked to find the perimeter of the parallelogram.
Applying the law of cosines for the triangle ABD,
x^{2} = a^{2} + b^{2}  2ab cos ∠BAD
Applying the cosine rule for the triangle ADC,
y^{2} = a^{2} + b^{2}  2ab cos ∠ADC
Adding the above two equations,
x^{2} + y^{2} = 2a^{2} + 2b^{2}  2ab (cos ∠BAD + cos ∠ADC) .... (1)
We know that any two adjacent angles of a parallelogram (it is a property of parallelograms) are supplementary. So
∠BAD + ∠ADC = 180^{o}
∠BAD = 180^{o}  ∠ADC
Applying cos on both sides,
cos ∠BAD = cos (180^{o}  ∠ADC) =  cos ∠ADC
Substiting this in (1),
x^{2} + y^{2} = 2a^{2} + 2b^{2}  2ab (  cos ∠ADC + cos ∠ADC)
x^{2} + y^{2} = 2a^{2} + 2b^{2}  2ab (0)
x^{2} + y^{2} = 2a^{2} + 2b^{2}
We have got the relation between the sides and diagonals of the parallelogram. Now, we will solve it for 'b' as we are not given the length of 'b'.
2b^{2} = x^{2} + y^{2}  2a^{2}
b^{2} = (x^{2} + y^{2}  2a^{2}) / 2
b = √ [(x^{2} + y^{2}  2a^{2}) / 2]
Now we know the sides of the parallelogram ('a' and 'b') and hence we can use the formula from the previous section to find its perimeter (P).
P = 2a + 2b
P = 2a + 2 √ [(x^{2} + y^{2}  2a^{2}) / 2]
P = 2a + √[2(x^{2} + y^{2}  2a^{2})
P = 2a + √(2x^{2} + 2y^{2}  4a^{2})
Perimeter of Parallelogram With Base, Height, and Angle
Let us consider a parallelogram with one of the sides given to be 'a', its corresponding height given to be 'h', and one of the vertex angles given to be 'θ'. Let us assume that the unknown side of the parallelogram be 'b'. We will first solve for 'b' and then find the perimeter.
Applying sin to the triangle BEC,
sin θ = h/b
b = h / sin θ
Thus, the perimeter (P) of the parallelogram is,
P = 2a + 2b
P = 2a + 2h / sin θ
Here θ doesn't need to be a specific angle of the parallelogram. It can be any vertex angle because any two adjacent angles of a parallelogram are supplementary and sin θ = sin (180  θ), for any θ.
Let us see the applications of the perimeter of parallelogram formulas in the following section.
Solved Examples on Perimeter of Parallelogram

Example 1: What is the perimeter of a parallelogram with the side length 8 in and diagonals 10 in and 12 in. Round your answer to two decimals.
Solution:
A side of the given parallelogram is, a = 8 in.
Let the other side be 'b' inches.
Its diagonals are, x = 10 in and y = 12 in.
Using the perimeter of a parallelogram formula using one side and diagonals, the perimeter (P) is
P = 2a + √(2x^{2} + 2y^{2}  4a^{2})
P = 2(8) + √(2(10)^{2} + 2(12)^{2}  4(8)^{2}) ≈ 31.23 inches
Answer: The perimeter of the given parallelogram = 31.23 inches.

Example 2: What is the perimeter of a parallelogram where one of its sides is 15 yards, its corresponding height is 20 yards, and one of the vertex angles is 30 degrees.
Solution:
One of the sides of the given parallelogram is, a = 15 yards.
Its height is, h = 20 yards.
One vertex angle is, θ = 30^{o}.
Its perimeter (P) is,
P = 2a + 2h / sin θ
P = 2(15) + (2 × 20) / (sin 30^{o}) = 110
Answer: The perimeter of the given parallelogram = 110 yards.
FAQs on Perimeter of Parallelogram
What Is the Perimeter of Parallelogram?
The perimeter of a parallelogram is the sum of all its sides. Thus, the perimeter of a parallelogram of sides 'a' and 'b' is a + a + b + b (or) 2a + 2b units.
What Is the Formula To Find the Perimeter of Parallelogram?
We have different formulas to find the perimeter of a parallelogram depending on the available information.
 The perimeter of a parallelogram whose adjacent sides are 'a' and 'b' is 2a + 2b.
 The perimeter of a parallelogram whose one of the sides is 'a' and whose diagonals are 'x' and 'y' is 2a + √(2x^{2} + 2y^{2}  4a^{2}).
 The perimeter of a parallelogram whose base is 'a', height is 'h', and one of the vertex angles is 'θ' is 2a + 2h / sin θ.
What Is the Perimeter of Parallelogram Using Diagonals?
We cannot find the perimeter of a parallelogram just using the diagonals, but if we know one of the sides as well, then we can find it. The perimeter of a parallelogram whose one of the sides is 'a' and whose diagonals are 'x' and 'y' is 2a + √(2x^{2} + 2y^{2}  4a^{2}).
What Is the Perimeter of Parallelogram Using Base and Height?
We cannot find the perimeter of a parallelogram when just base and height are given, but if we know a vertex angle as well, then we can find it. The perimeter of a parallelogram whose base is 'a', height is 'h', and one of the vertex angles is 'θ is 2a + 2h / sin θ.
How To Find the Perimeter of Parallelogram?
The perimeter of a parallelogram is obtained just by adding all its sides. Since the opposite sides of a parallelogram are equal, it is sufficient to know its two adjacent sides to find the perimeter. But we may not be given with two adjacent sides always, instead, we may be provided with some other information about it to find its perimeter.
 If adjacent sides of a parallelogram 'a' and 'b' are given then apply the formula 2a + 2b to find its perimeter.
 If one of the sides of a parallelogram is 'a' and its diagonals are 'x' and 'y' then apply the formula 2a + √(2x^{2} + 2y^{2}  4a^{2}) to find its perimeter.
 If the base of a parallelogram is 'a', its height is 'h', and one of its vertex angles is 'θ' then apply the formula 2a + 2h / sin θ to find its perimeter.
How To Find the Perimeter of a Parallelogram on Graph?
We know that the perimeter of a parallelogram whose adjacent sides are 'a' and 'b' is 2a + 2b. Thus, to find the perimeter of a parallelogram that is on a graph, first find the lengths of any two adjacent sides using the distance formula and then apply the formula 2a + 2b.
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