Parallelogram Formula

A parallelogram is a two-dimensional shape. It has four sides, in which two pairs of opposite sides are parallel. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure, while the sum of adjacent angles of a parallelogram is equal to 180 degrees. Before moving into the parallelogram formula let's have a look into its properties.

Properties of a Parallelogram

Few important properties of a parallelogram are listed below:

• The opposite sides are parallel and congruent.
• The interior opposite angles are congruent.
• The consecutive angles are supplementary.
• In case, if the measure of any one of the angles is a right angle, then all the other angles will be the right angle.
• The two diagonals bisect each other.
• Each diagonal bisects the parallelogram into two congruent triangles
• The sum of squares of all the sides of a parallelogram is equal to the sum of the square of its diagonals which is known as the parallelogram law.

What Is Parallelogram Formula?

Consider parallelogram ABCD as shown below. Let's understand the Parallelogram Formulas.

We have,           

• AB || CD, AC || BD                     
• AB = CD = b               
• BC = AD = a               
• h = height of the parallelogram
• DB = d\(_1\), AC = d\(_2\)  are digonals of the parallelogram

Area of a Parallelogram Formula

The area of a parallelogram is the region or surface occupied by a two-dimensional plane. Let's look into the parallelogram formula to calculate area.

Area of Parallelogram Formula Using Base and Height

• Area of parallelogram formula = Base × Height
• Area of parallelogram ABCD = b × h

Area of a Parallelogram Formula Without Height

• If the height of the parallelogram is not given, trigonometry can be used to find the area.
• Area of parallelogram formula = a b sin A = b a sin B

Area of a Parallelogram Using Diagonals

The area of any parallelogram can also be calculated using its diagonal lengths. Suppose, the diagonals d\(_1\) and d\(_2\) intersect each other at an angle x, then the area of the parallelogram is given by: 

• Area of parallelogram formula = ½ ×\({d_1} × {d_1} sin (x)\)
• Area of parallelogram ABCD = ½ × AC × BD sin (x)

Perimeter of a Parallelogram Formula

The perimeter of a parallelogram is the total distance of the boundaries of the parallelogram. To calculate the perimeter of a parallelogram, its length and breadth have to be known. Let the sides of the parallelogram be a and b. Keeping the property of a parallelogram in mind, i.e., the opposite sides are parallel and congruent. The perimeter of a parallelogram is given by:

• Perimeter = a + b + c + d (when a, b, c, d are representing 4 sides of a parallelogram)
• Since, side AB = CD = b, side BC = AD = a
• Perimeter of a parallelogram = 2 (a+b)

Example on Parallelogram Formula

Example 1: Using the parallelogram formula, find the area of the parallelogram with the base of 6 cm and height of 8 cm.

Solution:

Given:
Base, b = 6 cm
Height, h = 8 cm
We know that, using parallelogram formula
Area of Parallelogram formula = b × h square units
= 6 × 8 = 48 sq.cm

Answer: Therefore, the area of the parallelogram = 48 cm².

Example 2: Using the parallelogram formula, find the perimeter of a parallelogram with side measures 5 m and 7 m.

Solution:

Given:
a = 5 m
b = 7 m
We know that,
Perimeter of a Parallelogram = 2( a + b) units
= 2(5 + 7)
= 24 m

Answer: Thus, the perimeter of the parallelogram is 24 m.

Example 3: The area of a playground that is in the shape of a parallelogram is 25000 in², with one side measuring 250 in. Find the corresponding altitude using the area of the parallelogram formula. 

Solution:
Area of the playground = 25000 in²
Side of a playground = 250 in
Using Area of Parallelogram formula = b × h square units
Corresponding altitude = 25000/250 = 100 in

Answer: Corresponding altitude of the playground measures 100 inches.