Perimeter of a Parallelogram: Properties, Formula & Examples

Perimeter of a parallelogram is equivalent to the sum of all sides of the parallelogram. It is a type of quadrilateral that has four sides. When the opposite sides of a quadrilateral are parallel and of equal length, it is termed as parallelogram. 

• Perimeter of a parallelogram is the distance covered along the boundary.
• The perimeter is calculated when sides and diagonals are known.
• The opposite angles of the figure are equal.
• Diagonals bisect each other at 90 degrees.
• Θ is the angle formed at the vertex of the parallelogram.
• There is no line of symmetry in the case of the given figure.
• The rhombus, rectangle, and square are all examples of parallelograms.
• Perimeter of a parallelogram is equal to the perimeter of a rectangle.

Read More: Area of Parallelogram

Key Terms: Perimeter of a Parallelogram, Parallelogram, Rectangle, Rhombus, Square, Sides, Diagonals, Quadrilateral, Height

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Perimeter of a Parallelogram

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The perimeter of a parallelogram is equal to the sum of all its sides since it is the length of its outline. Parallelogram is a type of quadrilateral, which has four sides. 

• The pair of opposite sides are parallel and equal to each other in a parallelogram. 
• The length, width and height measure the area and perimeter of the parallelogram.

The perimeter of a parallelogram is calculated in the following cases:

• When two adjacent sides are known.
• When one side and diagonals are known.
• When the height, base and any angle are given.

Perimeter of a Parallelogram

Read More: Difference between Square and Rhombus

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How to find Perimeter of a Parallelogrm?

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There are three cases available to calculate the perimeter of a parallelogram which are discussed in below section:

Perimeter of a Parallelogram when sides are given

In case of a parallelogram, opposite sides are equal. Let’s consider a parallelogram with two adjacent sides named a and b.Then Perimeter of parallelogram is a+b+c+d or 2a + 2b= 2(a+b). Therefore, P = 2(a+b).

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Perimeter of a Parallelogram when one side and diagonals are given

Let us consider a parallelogram with sides a & b and diagonals y & x. Now apply the law of cosines rule in triangle ABD,

• x² = a² + b² – 2ab.cos ∠BAD ---(1)
• Also apply cosine rule in triangle ADC,
• y² = a² + b² – 2ab.cos ∠ADC ---(2)
• Now, By adding equation 1 and 2 then we get;
• x² + y² = 2a² + 2b² – 2ab (cos ∠BAD + cos ∠ADC ) ---(3)
• We know that the two adjacent angles of Parallelogram are 180^o.
• So, ∠BAD + ∠ADC = 180^o
• ∠BAD = 1800 - ∠ADC
• By applying cos on both sides
• Cos ∠BAD = cos (180^o - ∠ADC)
• Cos ∠BAD = - cos ∠ADC ---(4)
• Now, put equation number 4 in equation 3, we get
• x² + y² = 2a² + 2b² – 2ab (- cos ∠ADC + cos ∠ADC) 
• x² + y² = 2a² + 2b² – 2ab(0)
• x² + y² = 2a² + 2b²
• For length ‘b’
• 2b² = x² + y² - 2a²
• b² = (x² + y² – 2a²) /2 
• b = √ [(x² + y² – 2a²) /2 ]
• Now the formula for perimeter (P)is:
• P = 2a +2b
• P = 2a + 2 √ [(x² + y² – 2a²) /2 ]
• P = 2a + √ [(x² + y² – 2a²)

P = 2a + √ [(2x² +2y² – 4a²)

Perimeter of a Parallelogram when the height, base and angle are given

Let consider the parallelogram with one of the sides a and its corresponding to height ‘h’ and with vertex angle θ. Assume the unknown sides of parallelogram be ‘b’.

• By applying sin rule in triangle BEC,
• Sin θ = h/b
•  b= h/ sin θ
• therefore, the perimeter of parallelogram is:
• P = 2a + 2b

P = 2a + 2h/ sin θ

Read More: Perimeter of a Trapezoid

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Perimeter of a Parallelogram Formula

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There are different formulas to calculate the perimeter of a parallelogram using different components. Let a and b be the two sides of the parallelogram. Θ is the angle formed between the vertex of parallelogram and h is the height of the parallelogram.

• Perimeter of a Parallelogram when sides are given

P = 2(a+b)

• Perimeter of a Parallelogram when one side and diagonals are given

P = 2a + √ [(2x² +2y² – 4a²)

• Perimeter of a Parallelogram when the height, base and angle are given

P = 2a + 2h/ sin θ

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Things to Remember

• The perimeter of a parallelogram is the sum of all lengths enclosed by the boundary.
• A parallelogram is a four-sided polygon with two parallel sides that also has four edges.
• If one of the angles in a parallelogram is a right angle, then all four angles must be right angles.
• The perimeter of a parallelogram is also equivalent to the perimeter of a rectangle.
• The length of the opposite sides of a parallelogram is the same because they are equal in length.

Read More: Areas of Parallelograms and Triangle