Area of a Parallelogram: Simple Formulas, Steps, and Examples

Area of Parallelogram

Area of a Parallelogram: In geometry, a parallelogram is a simple quadrilateral featuring two pairs of parallel, equal-length opposite sides and equal opposite angles. It encompasses various shapes, including squares, rectangles, and rhombi. You likely encounter these shapes daily in tabletops, windows, and architectural designs. In this guide, we will explore the definition of a parallelogram, its area formulas, and practical calculation examples.

What is Parallelogram?

A parallelogram is essentially a polygon with four sides where opposite sides are parallel. Beyond this fundamental structure, parallelograms possess several distinct geometric properties, including:

• Their opposite sides are parallel.
• Their Opposite sides are equal.
• Their Opposite angles are equal.
• The diagonals bisect each other.
• Their interior angle is supplementary to each other.

What is the Area of Parallelogram? 

The area of a parallelogram refers to the total two-dimensional space enclosed within its boundaries. Mathematically, this area is determined by calculating the product of the parallelogram's base and its corresponding perpendicular height.

Area of Parallelogram Formula

To find the area of a parallelogram, you must multiply the length of its base by its vertical height. Note that the height must be measured perpendicular to the base. The primary formula is as follows:

Area of Parallelogram = b × h square units

Where: 

b = the length of the base

h = the perpendicular height or altitude

Calculation of Area of Parallelogram 

While the base-height method is standard, you can also determine the area of a parallelogram using its diagonals and their intersecting angles, or by using the lengths of two adjacent sides and the angle between them. Effectively, there are three primary approaches to calculating this area.

Calculation of Area of Parallelogram
Parameters	Formula
Using height and base	Area = Base × Height
Using  diagonals	Area = ab sin (θ)
Using the length of the sides	Area = ½ × d1 × d2 sin (x)

Let's look at some solved examples:

• b = base of parallelogram
• h = height of a parallelogram
• a = the side of the parallelogram
• θ = angle between the sides of the parallelogram
• d1 = diagonal of the parallelogram
• d2 = diagonal of the parallelogram
• x = any angle that is between the intersection point of the diagonals.

Area of Parallelogram Examples

Question 1. The base of a parallelogram is three times its height. If the total area is 243 cm², find the base and height.

Solution: Let the height of the parallelogram = x cm.

Therefore, the base of the parallelogram = 3x cm.

Area = 243 cm²

Using the formula: Area = base × height

243 = 3x × x

⇒ 3x² = 243

⇒ x² = 81

⇒ x = 9

Since the base is 3x, then 3 × 9 = 27.

Therefore, the base of the parallelogram is 27 cm, and the height is 9 cm.

Question 2. A parallelogram has adjacent sides of 15 cm and 10 cm. If the distance between its shorter sides is 9 cm, find the distance between its longer sides.

Solution: Sides are 15 cm and 10 cm.

Distance between shorter sides (the height relative to the 10 cm side) = 9 cm.

Area of parallelogram = base × height = 10 × 9 = 90 cm².

Now, use the same area for the longer base: 90 = 15 × h

⇒ h = 90 / 15

⇒ h = 6 cm

The distance between the longer sides is 6 cm.

Therefore, the distance between its longer sides is 6 cm. 

Question 3. Calculate the area of a parallelogram-shaped roof with a base of 10 inches and an altitude of 5 inches.

Solution: Using the formula: Area = b × h

Area = 10 × 5 = 50 in².

The area of the roof is 50 in².

Question 4: The area of a parallelogram is 300 cm². Its height is twice its base. Find the dimensions.

Solution: Area = 300 cm², height (h) = 2b.

Area = b × h

300 = b × (2b)

300 = 2b²

b² = 150

b = √150 ≈ 12.25 cm

The base is approximately 12.25 cm.  

The base of the parallelogram is approximately 12.25 cm.

Therefore, the height is approximately 24.5 cm.

Question 5. If the angle between two adjacent sides of 9 cm and 12 cm is 90°, find the area.

Solution: a = 9 cm, b = 12 cm, θ = 90°.

Using trigonometry: Area = a × b × sin(θ)

Area = 9 × 12 × sin(90°)

Area = 108 × 1

Area = 108 cm²

The area is 108 cm².