Perimeter of Parallelogram

A parallelogram has some properties similar to that of a rectangle. We also consider a parallelogram as a slanting rectangle. However, a parallelogram is slightly different from a rectangle in terms of some properties. So its perimeter is found differently. Let us learn how to find the perimeter of a parallelogram here.

What is the Perimeter of a Parallelogram

The perimeter of a parallelogram is the total distance covered around its edge. In other words, the perimeter of a parallelogram is the total sum of its four sides.

Since the perimeter measures length or distance, its unit is always linear. e.g., as cm, m, inch, ft, or yd.

Formulas

There are different ways to find the perimeter of a parallelogram based on the information given.

Using Side Lengths

Let ‘a’ and ‘b’ be the sides of a parallelogram. We know that the perimeter of a parallelogram is the sum of all its sides. Also, the opposite sides of a parallelogram are parallel and equal to each other. Thus, the perimeter ‘P’ of a parallelogram is:

P = a + b + a + b units

P = 2a + 2b

P = 2(a + b)

Thus, the standard formula to calculate the perimeter of a parallelogram is given below:

Let us solve an example to clear your concept better.

Other Ways to Find the Perimeter of a Parallelogram

We might not always know all the sides of a parallelogram. The perimeter of a parallelogram can be calculated when other information is given, as in the following cases –

• When one side and diagonals are given
• When base, height, and any angle are given

With One Side and Diagonals

Consider a parallelogram ABCD with sides ‘a’ & ‘b’ and diagonals ‘x’ & ‘y’. Here, ‘a’, ‘x’ and ‘y’ are known. The value of ‘b’ is unknown. We have to calculate the perimeter of the parallelogram.

Derivation

Applying the law of cosines for the ΔABC,

x² = a² + b² – 2ab cos∠ ABC

Applying the cosine rule for the ΔBAD

y² = a² + b² – 2ab cos∠ BAD

Adding the above two equations,

x² + y² = 2a² + 2b² – 2ab (cos∠ABC + cos∠BAD) …. (1)

Since, any 2 adjacent angles of a parallelogram add up to 180° (supplementary angles), so,

∠ABC + ∠BAD = 180°

or, ∠ABC = 180° – ∠BAD

Applying cos on both sides,

cos∠ABC = cos(180° – ∠BAD) = – cos∠BAD

⇒ x² + y² = 2a² + 2b² – 2ab ( – cos∠BAD + cos∠BAD) , from …. (1)

⇒ x² + y² = 2a² + 2b² – 2ab (0)

⇒ x² + y² = 2a² + 2b² …. (the relation between the sides and diagonals)

⇒ 2b² = x² + y² – 2a²

⇒ b² = (x² + y² – 2a²)/2

⇒ b = √[(x² + y² – 2a²)/2]

as we know,

P = 2(a + b)

⇒ P = 2a + 2 √[(x² + y² – 2a²)/2]

⇒ P = 2a + √[2(x² + y² – 2a²)

⇒ P = 2a + √(2x² + 2y² – 4a²)

The formula to calculate the perimeter of a parallelogram with one side and the 2 diagonals is given below:

With Base, Height and Angle

Let us find the perimeter of parallelogram ABCD with base, height and any vertex angle.

Derivation

In ΔCED,

sinθ = h/a

a = h/sinθ

∴The perimeter (P) of the parallelogram is,

P = 2a + 2b

⇒ P = 2(h/sinθ) + 2b

Here θ any vertex angle because any two adjacent angles of a parallelogram are supplementary. So,

sinθ = sin(180° – θ), for any θ

The  formula to calculate the perimeter of a parallelogram with base, height, and any vertex angle is given below:

Let us solve an example to understand the above formula better.

Area and Perimeter of a Parallelogram

Let us find the relationship between the area and perimeter of a parallelogram.

We know,

Area of a parallelogram (A) = b x h square units ……(1), here b = base, h = height

And,

Perimeter of a parallelogram (P) = 2(a + b) units, here a & b are any 2 adjacent sides

Now, value of b in terms of P is

⇒ P/2 = a + b

b = (P/2) – a

A = ((P/2) – a)h, Substitute the value of b from (1)

∴A = ((P/2) – a)h square units

Let us solve some examples to understand the relationship.