Parallelogram: Definition, Properties, Types, Formulas & Examples

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Its opposite sides are also equal, opposite angles are equal, and its diagonals bisect each other.

The main parallelogram formulas are:

• Area = base × height

• Perimeter = 2 × (base + side)

Understanding the definition, properties, types, formulas and examples of a parallelogram helps in solving geometry questions involving sides, angles, area and perimeter.

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What Is a Parallelogram?

A parallelogram is a four-sided polygon (quadrilateral) with two pairs of opposite parallel sides.

For a parallelogram ABCD:

• AB ∥ CD

• AD ∥ BC

• AB = CD

• AD = BC

The opposite angles are equal, while adjacent angles add up to 180°.

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Parallelogram Diagram

Consider parallelogram ABCD with diagonals AC and BD intersecting at O.

The diagram can be used to identify its:

• Four sides

• Two pairs of parallel sides

• Four interior angles

• Two diagonals

• Equal opposite sides

• Bisecting diagonals

Parallelogram meaning: A parallelogram is a quadrilateral whose opposite sides are parallel and equal.

Definition of Parallelogram

The definition of a parallelogram is:

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

If ABCD is a parallelogram:

AB ∥ CD and AD ∥ BC

Because opposite sides are parallel, several other properties follow, including equal opposite sides and equal opposite angles.

Properties of a Parallelogram

The important parallelogram properties are:

1. Opposite sides are parallel.

2. Opposite sides are equal.

3. Opposite angles are equal.

4. Adjacent angles are supplementary.

5. The sum of all four interior angles is 360°.

6. The diagonals bisect each other.

7. Each diagonal divides the parallelogram into two congruent triangles.

For parallelogram ABCD:

• AB ∥ CD

• AD ∥ BC

• AB = CD

• AD = BC

• ∠A = ∠C

• ∠B = ∠D

• ∠A + ∠B = 180°

• ∠B + ∠C = 180°

• AO = OC

• BO = OD

Important Point About the Diagonals

The diagonals of a general parallelogram bisect each other, but they are not necessarily equal in length.

Equal diagonals are a property of a rectangle and square, while perpendicular diagonals are a property of a rhombus and square.

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Types of Parallelograms

The main special types of parallelograms are:

1. Rectangle

2. Rhombus

3. Square

A general parallelogram does not necessarily have equal diagonals or right angles.

1. Rectangle

A rectangle is a parallelogram with four right angles.

Its properties include:

• Opposite sides are equal and parallel.

• All four angles are 90°.

• Diagonals are equal.

• Diagonals bisect each other.

For rectangle ABCD:

• AB ∥ CD

• AD ∥ BC

• AB = CD

• AD = BC

• ∠A = ∠B = ∠C = ∠D = 90°

• AC = BD

2. Rhombus

A rhombus is a parallelogram in which all four sides are equal.

Its properties include:

• All four sides are equal.

• Opposite sides are parallel.

• Opposite angles are equal.

• Diagonals bisect each other.

• Diagonals are perpendicular.

• Each diagonal bisects a pair of opposite angles.

For rhombus ABCD:

• AB = BC = CD = DA

• AB ∥ CD

• AD ∥ BC

• ∠A = ∠C

• ∠B = ∠D

• AC ⟂ BD

Note: The diagonals of a rhombus are not generally equal. They are equal when the rhombus is a square.

3. Square

A square is a parallelogram with four equal sides and four right angles.

Its properties include:

• All four sides are equal.

• Opposite sides are parallel.

• All four angles are 90°.

• Diagonals are equal.

• Diagonals bisect each other.

• Diagonals are perpendicular.

For square ABCD:

• AB = BC = CD = DA

• AB ∥ CD

• AD ∥ BC

• ∠A = ∠B = ∠C = ∠D = 90°

• AC = BD

• AC ⟂ BD

General Parallelogram

A general parallelogram has two pairs of parallel opposite sides. Its opposite sides and opposite angles are equal, and its diagonals bisect each other.

However, its diagonals are not necessarily equal or perpendicular.

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Parallelogram Formulas

The two basic formulas used for a parallelogram are the area formula and perimeter formula.

Area of a Parallelogram

The area of a parallelogram is the product of its base and perpendicular height.

Formula:

Area = base × height

or

A = b × h

Where:

• A = area

• b = base

• h = perpendicular height

Example

If the base is 15 cm and the perpendicular height is 8 cm:

Area = 15 × 8

Area = 120 cm²

Important: Base and Height

The height of a parallelogram is the perpendicular distance between its parallel sides.

It is not necessarily the length of the slanting side.

Perimeter of a Parallelogram

The perimeter is the total distance around the parallelogram.

If the adjacent sides are a and b:

Perimeter = 2(a + b)

Example

If the adjacent sides are 15 cm and 6 cm:

Perimeter = 2(15 + 6)

Perimeter = 42 cm

Therefore, the perimeter is 42 cm.

Area and Perimeter of a Parallelogram

Here, a and b are adjacent sides, while h is the perpendicular height corresponding to the selected base.

Parallelogram Examples

Example 1: Find the Area and Perimeter

A parallelogram has a base of 15 cm, a perpendicular height of 8 cm, and an adjacent side of 6 cm. Find its area and perimeter.

Solution:

Area = base × height

= 15 × 8

= 120 cm²

Perimeter = 2(base + side)

= 2(15 + 6)

= 2 × 21

= 42 cm

Answer:

• Area = 120 cm²

• Perimeter = 42 cm

Example 2: Find the Angles of a Parallelogram

In parallelogram ABCD, ∠A + ∠C = 70°. Find all four angles.

Opposite angles of a parallelogram are equal.

Therefore:

∠A = ∠C

Given:

∠A + ∠C = 70°

Therefore:

2∠A = 70°

∠A = 35°

So:

∠C = 35°

Adjacent angles of a parallelogram are supplementary.

Therefore:

∠A + ∠B = 180°

35° + ∠B = 180°

∠B = 145°

Since opposite angles are equal:

∠D = 145°

Therefore:

• ∠A = 35°

• ∠B = 145°

• ∠C = 35°

• ∠D = 145°

Parallelogram: Quick Revision

Remember these points:

• A parallelogram is a quadrilateral.
• Both pairs of opposite sides are parallel.
• Opposite sides are equal.
• Opposite angles are equal.
• Adjacent angles add up to 180°.
• The sum of all four angles is 360°.
• Diagonals bisect each other.
• Area = base × perpendicular height.
• Perimeter = 2(a + b).
• A rectangle, rhombus and square are special types of parallelograms.

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