# Perimeter of a Parallelogram

**Perimeter of a Parallelogram:**

h: height

For any parallelogram the following statements are true:

ab = dc

ad = bc

The perimeter of a parallelogram is therefore given by:

*P = 2 (ad + ab)*

If the length of ad or bc is not given, it can be calculated by using the following formula:

*ad = bc = h/cos(**∠**ecb)*

where ∠ecb is the angle measured in degree or radian

∠ecb can be calculated if the length of eb is given:

∠ecb = tan^{-1}(eb/h)

**Example 1:** Find the area and perimeter of the parallelogram, whose base is 18 cm and the height is 4 cm, also the angle between the base and the side is 130^{o }and 50^{o}.

__Solution__:

Given that:

base = dc = ab = 18 cm

h = 4 cm

Area of parallelogram:

*A = b · h*

*A = 18 · 4*

*A = 72*

Thus, area of parallelogram = 72 cm^{2}

Perimeter of parallelogram:

P = 2(dc + ad)

Since the length of ad is not known, we need to determine it first using *ad = bc = h/cos(**∠**ecb)*

Inserting* **∠**ecb = 150*^{ o}* – 90*^{ o}* = 60*^{o}

*ad = bc = 4/cos(**∠**60*^{ o}*)*

*ad = bc = 8 cm*

*P = 2 · (18 + 8) = 2 ·26 = 52 cm*

**Example 2:** The area of the parallelogram ABCD is 54 m^{2} and its perimeter is 34 m. What are the dimensions of the parallelogram?

__Solution__:

**Step 1:** The area of the parallelogram ABCD = base × height = BC × AE = 54 m^{2} [Given, area of ABCD = 54 m^{2}]

**Step 2:** From the figure above, we can see that the height of the parallelogram is AE = 6 m

**Step 3:** The base length of the parallelogram is *BC = 54 · AE = 546 = 9 m*[Substitute AE = 6.]

**Step 4:** The perimeter of the parallelogram = *AB + BC + AB + BC* [From the figure, since CD = AB and AD = BC.]

**Step 5:** [Substitute the value of the perimeter of the parallelogram.] (2 × AB) + (2 × BC) = 34

**Step 6:** [Distributive property.]2 × (AB + BC) = 34

**Step 7:** [Divide each side by 2.]AB + BC = 17

**Step 8:** [Substitute base length BC =9 m.]AB + 9 = 17

**Step 9:** [Subtract 9 from both the sides.]AB = 17 - 9 = 8 m

**Step 10:** The dimensions of the parallelogram are l = 9 m and b = 8 m.

**Example 3:** Find the perimeter of a parallelogram whose slant height is 24, height is 22 and breadth is 26, all units are in measured in mm.

__Solution__:

Given that:

a = slant height of the parallelogram

b= breadth of the parallelogram

Perimeter of a parallelogram = *2*(24+26) = 2*50 = 100 mm*

**Example 4:** Find the perimeter of the parallelogram ABCD.

__Solution__:

Perimeter of the parallelogram ABCD = AB + BC + CD + AD

AB = DC = 4 cm and AD = BC = 5 cm

*Perimeter = 4 + 5 + 4 + 5 = 18 cm*

Online Perimeter Calculator