# 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.

Find the perimeter of a parallelogram whose base and side lengths are 6 cm and 9 cm, respectively.

Solution:

As we know,
P = 2(a + b), here a = 6 cm, b = 9 cm
= 2(6 + 9)
= 2 × 15
= 30 cm

### 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

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,

or, ∠ABC = 180° – ∠BAD

Applying cos on both sides,

⇒ 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:

Finding the perimeter of a parallelogram when ONE SIDE and DIAGONALS are known

Find the perimeter of a parallelogram with the side length 7 cm and diagonals 9 cm and 11 cm. Round your answer to two decimals.

Solution:

As we know,
P = 2a + √(2x2 + 2y2 – 4a2) , here a = 7 cm, x = 9 cm, y = 11 cm
= 2 × 7 + √(2 × 92 + 2 × 112 – 4 × 72)
= 14 + √(162 + 242 – 196)
= 14 + √208
= 28.42 cm

#### 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.

Finding the perimeter of a parallelogram when BASE, HEIGHT, and VERTEX ANGLE are known

Find the perimeter of a parallelogram where one of its sides is 12 ft, its corresponding height is 8 ft, and one of the vertex angles is 30 degrees.

Solution:

As we know,
P = 2(h/sinθ) + 2b , here h = 8 ft, b = 12 ft, θ = 30°
= 2(8/sin 30°) + 2 × 12
= 2(8 ÷ 1/2) + 24 , since sin 30° = 1/2
= 32 + 24
= 56 ft

## 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.

Finding the perimeter of a parallelogram when ONE SIDE, HEIGHT, and AREA are known

Find the perimeter of a parallelogram with a height of 6 cm and an adjacent side of 8 (which is not the base) cm whose area is 36 sq. cm.

Solution:

As we know,
A = ((P/2) – a)h, here a = 8 cm, h = 6 cm, A = 36 sq. cm
∴ P = 2((A/h) + a)
= 2(36/6 + 8)
= 28 cm

Finding the length of one side of a parallelogram when BASE and PERIMETER are known

Find the length of another side of the parallelogram whose base is 6 cm and the perimeter is 42 cm.

Solution:

As we know,
a = P/2 – b , here a = unknown side, b = 6 cm, P = 42 cm
= 42/2 – 6
= 21 – 6 = 15 cm