Parallelogram Definition, Formulas, Types and Examples

What is Parallelogram?

Parallelogram meaning - A parallelogram is a two-dimensional geometrical figure with four sides where the opposite sides are parallel and equal in length. If you are wondering “what is a parallelogram” in simple terms, think of it as a slanted rectangle that still follows special rules of geometry. Understanding parallelogram definition, properties, and formulas makes it easier to solve mathematical problems related to geometry, trigonometry, and even real-life applications in physics and engineering.

Learning about a parallelogram also helps in building problem-solving skills, as it connects to algebra, coordinate geometry, and vectors. Students often ask “what is parallelogram in geometry?” or “what is a parallelogram used for?” — the answer lies in its wide use in both academics and practical life.

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Definition of Parallelogram

A parallelogram definition in geometry states that it is a quadrilateral in which both pairs of opposite sides are parallel and equal in length.

For example, ABCD is a parallelogram.

Therefore, we can say,

• AB and CD are parallel and equal

• AD and BC are parallel and equal

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Properties of Parallelogram

A parallelogram has its own set of unique properties. To truly understand what is a parallelogram, one must explore its key properties. These properties hold for all types of parallelogram and are used in solving many geometry problems.

Key Parallelogram Properties:

• Opposite sides are equal and parallel

• Opposite angles are equal

• Adjacent angles are supplementary (sum is 180°)

• The sum of all angles is 360°

• Diagonals bisect each other

• Each diagonal divides the parallelogram into two congruent triangles

Let’s explain these properties with the help of the following diagram.

In the above parallelogram ABCD, we can relate the following properties:

• Opposite sides are parallel, which means AB || DC and AD || BC

• Opposite sides are equal, which means AB = DC and AD = BC

• Opposite angles are equal, which means ∠A = ∠C and ∠B = ∠D

• Adjacent angles are supplementary which means ∠D + ∠C = 180° and ∠A + ∠B = 180°

• The sum of all angles is 360° which means ∠A + ∠B + ∠C + ∠D = 360°

• Diagonals bisect each other, which means AE = CE and DE = BE

• Each diagonal divides the parallelogram into two congruent triangles, which means that triangles ABD and CBD are congruent and triangles ADC and ABC are congruent. 

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Types of Parallelograms

The types of parallelogram are classified based on their sides, angles, and diagonals. Common types include:

Rectangle

It is a type of parallelogram in which all the internal angles are 90 degrees, and opposite sides are equal and parallel. The diagonals are of equal length.

In the rectangle ABCD, we can relate the following properties:

• Two pairs of parallel sides: AB || DC and AD || BC

• Four right angles: ∠A = ∠B = ∠C = ∠D = 90∘

• Opposite sides of equal lengths: AB = DC and AD = BC

• Two equal diagonals: AC = BD

• Diagonals bisect each other.

Rhombus

It is a type of parallelogram in which all sides are equal and opposite sides are parallel. The internal angles are not right angles. The diagonals are equal in length and bisect each other at right angles.

In the rhombus ABCD, we can relate the following properties:

• Two pairs of parallel sides: AB || DC and AD || BC

• Four equal sides: AB = BC = CD = DA

• Equal opposite angles: ∠A = ∠C and ∠B = ∠D

• Diagonals that are perpendicular to each other: BD⊥AC

• Diagonals bisect each other: BM = DM and AM = CM 

Square

This type of parallelogram has four equal sides, and all internal angles are 90 degrees. The opposite pair of sides is parallel. The diagonals are equal and bisect each other at right angles.

In the square ABCD, we can relate the following properties:

• Two pairs of parallel sides: AB || DC and AD || BC

• Four equal sides: AB = BC = CD = DA

• Four right angles: ∠A = ∠B = ∠C = ∠D = 90∘

• Diagonals are equal: AC = BD

• Diagonals are perpendicular to each other: BD⊥AC

• Diagonals bisect each other: AO = CO and DO = BO

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General Parallelogram

It is the typical parallelogram with no right angles, and opposite sides are equal and parallel. The diagonals may or may not be equal in length, but they bisect each other.

In the general parallelogram ABCD, we can relate the following properties:

• Two pairs of parallel sides: AB || DC and AD || BC

• Opposite sides are equal:  AB = DC and AD = BC

• Equal opposite angles: ∠A = ∠C and ∠B = ∠D

• Diagonals bisect each other: AO = CO and DO = BO

Parallelogram Formulas

Before solving problems, it is important to understand the parallelogram formulas used to calculate its area and perimeter. These formulas are simple yet highly useful in geometry, trigonometry, and even real-world applications like land measurement, construction, and physics. By learning the correct formulas, students can easily solve questions based on different types of parallelogram.

Area of Parallelogram

The area of a parallelogram is calculated by multiplying the length of its base and its height (the perpendicular distance between the base and the opposite parallel side).

So, the formula for the area of a parallelogram is:

Area (A) = Base (b) X Height (h)

Perimeter of Parallelogram

The perimeter of a parallelogram is the total length of all four sides. The formula for the perimeter of a parallelogram is:

Perimeter (P) = 2 (sum of adjacent sides)

                       = 2 (a + b)

Where a and b are the lengths of adjacent sides.

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Parallelogram Examples 

1. Find the area and perimeter of a parallelogram with base 15 cm, height 8 cm, and side length 6 cm.

Solution:

• Area = base × height = 15 × 8 = 120 cm²

• Perimeter = 2 × (15 + 6) = 2 × 21 = 42 cm 

2. In a parallelogram ABCD, the sum of the two opposite angles ∠A and ∠C is 70 degrees. Find all four angles of the parallelogram.

Solution:

In the parallelogram, the opposite angles are equal.

Therefore, ∠A = ∠C

Again, ∠A + ∠C = 70

Or, ∠A + ∠A = 70

Or, 2∠A = 70

Or, ∠A = 35

Therefore, ∠C = 35 degrees

Again, the sum of adjacent angles in the parallelogram is 180 degrees.

So, ∠A + ∠D = 180 degrees

Or, ∠D = 180 - ∠A

Or, ∠D = 180 – 35 = 145

Again, in the parallelogram, ∠D = ∠B

So, we can say ∠B = 145 degrees

Therefore, the measurements of all four angles of the parallelogram are:

∠A = 35 degrees, ∠B = 145 degrees, ∠C = 35 degrees, and ∠D = 145 degrees

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Importance of Learning About Parallelograms

Studying parallelograms and their properties helps strengthen your understanding of geometry, vectors, and algebra. In higher mathematics, concepts from parallelograms apply to different topics, which include:

• Coordinate geometry

• Trigonometry

• Vector mathematics

• Physics and engineering problems

Understanding the parallelogram as a special geometrical shape clarifies mathematical concepts and real-world applications. Knowing how to use parallelogram formulas, recognise properties, and identify types of parallelograms helps in problem-solving in competitive exams quickly and accurately.

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