Parallelogram: Definition, Properties & Formulas

Definition: A parallelogram is a four-sided closed figure with two pairs of parallel sides, where opposite sides are equal in length and opposite angles are equal in measure.

What is a Parallelogram?

A parallelogram is a type of quadrilateral defined by a single condition: both pairs of opposite sides must be parallel. Once that condition is met, several other properties follow automatically — opposite sides become equal in length, opposite angles become equal in measure, and the diagonals bisect each other.

In a parallelogram ABCD, the vertices are labelled in order around the shape. Sides AB and CD form one pair of parallel sides, and sides AD and BC form the other. The notation AB ∥ CD means side AB is parallel to side CD. Similarly, AB = CD means the two sides are equal in length.

The diagonals AC and BD connect opposite vertices and intersect at a single interior point, usually labelled O.

Properties of a Parallelogram

Five main properties define every parallelogram:

1. Opposite sides are equal and parallel. AB = CD and AD = BC; AB ∥ CD and AD ∥ BC.

2. Opposite angles are equal. ∠A = ∠C and ∠B = ∠D.

3. Consecutive (adjacent) angles are supplementary. Any two angles sharing a side add to 180°. So ∠A + ∠B = 180°, ∠B + ∠C = 180°, and so on.

4. Diagonals bisect each other. The diagonals AC and BD intersect at point O, where AO = OC and BO = OD.

5. Each diagonal divides the parallelogram into two congruent triangles. Diagonal AC splits the parallelogram into triangles ABC and CDA, which are congruent in shape and size.

The sum of all four interior angles equals 360°, as it does for any quadrilateral.

Conditions to Prove a Quadrilateral is a Parallelogram

Not every quadrilateral is a parallelogram. To prove that a given quadrilateral is one, only one of the following five conditions needs to hold:

1. Both pairs of opposite sides are parallel.

2. Both pairs of opposite sides are equal in length.

3. Both pairs of opposite angles are equal in measure.

4. The diagonals bisect each other.

5. One pair of opposite sides is both equal and parallel.

Each condition is sufficient on its own. Meeting any one of them guarantees that the remaining properties follow.

Types of Parallelograms

Parallelograms are classified into four types based on their side lengths and angles.

Rectangle

A rectangle is a parallelogram with all four interior angles equal to 90°. Opposite sides are equal in length, and the diagonals are equal in length (in addition to bisecting each other).

Rhombus

A rhombus is a parallelogram with all four sides of equal length. Its diagonals bisect each other at right angles, and they also bisect the interior angles.

Square

A square is a parallelogram that is both a rectangle and a rhombus — all four sides are equal, and all four angles are 90°. The diagonals are equal in length, perpendicular to each other, and bisect each other.

Rhomboid

A rhomboid is a parallelogram with adjacent sides of unequal lengths and no right angles. It is the "general" parallelogram — neither a rectangle nor a rhombus. The term is less common in modern mathematics but appears in older geometry texts and in biology (e.g., rhomboid muscles, rhomboid leaf shapes).

Comparison Table

Parallelogram Formulas

A parallelogram has standard formulas for area, perimeter, height, and diagonal lengths. The correct formula depends on which measurements are available.

Area of a Parallelogram

Three formulas exist for the area of a parallelogram. Use the one that matches the inputs you have.

Formula 1 — When base and height are known:

Area = b × h

This is the most common formula. Use it when the perpendicular height is given.

Formula 2 — When two adjacent sides and the included angle are known:

Area = a × b × sin(θ)

Use this when the perpendicular height is not given but the angle between adjacent sides is known.

Formula 3 — When both diagonals and the angle between them are known:

Area = ½ × d₁ × d₂ × sin(α)

This is the correct general form. The shorter version ½ × d₁ × d₂ (without sin α) applies only when the diagonals are perpendicular — that is, for a rhombus or a square. For a general parallelogram, the sin(α) term must be included.

Perimeter of a Parallelogram

Perimeter = 2(a + b)

Since opposite sides are equal, only two distinct side lengths are needed.

Height of a Parallelogram

When the area and base are known, the perpendicular height can be calculated as:

h = Area / b

This is useful for working backwards — for example, when an area is given and the perpendicular distance between two parallel sides needs to be found.

Length of Diagonals

Diagonal lengths can be calculated using the law of cosines applied to the triangles formed by the diagonals:

d₁² = a² + b² − 2ab cos(θ)d₂² = a² + b² + 2ab cos(θ)

The sum of the squares of both diagonals equals 2(a² + b²) — a result known as the parallelogram law.

The Parallelogram Law

The parallelogram law is an identity that connects the sides of a parallelogram to its diagonals.

The Parallelogram Law: The sum of the squares of all four sides equals the sum of the squares of the two diagonals.

2(a² + b²) = d₁² + d₂²

This identity is used in vector geometry and physics. In the parallelogram law of vector addition, two vectors are represented as adjacent sides of a parallelogram, and their resultant is the diagonal drawn from their common starting point.

Worked Example

Problem: A parallelogram ABCD has adjacent sides AB = 10 cm and BC = 6 cm. The angle ∠ABC = 30°. Find: (a) The perimeter (b) The area using the angle formula (c) The perpendicular height from side AB to side CD

Given: a = 10 cm, b = 6 cm, θ = 30°

Solution:

(a) Perimeter: Perimeter = 2(a + b) Perimeter = 2(10 + 6) Perimeter = 32 cm

(b) Area using two sides and the included angle: Area = a × b × sin(θ) Area = 10 × 6 × sin(30°) Area = 60 × 0.5 Area = 30 cm²

(c) Perpendicular height (using base AB = 10 cm): h = Area / b h = 30 / 10 h = 3 cm

This matches the alternative calculation h = BC × sin(θ) = 6 × sin(30°) = 3 cm.

Parallelogram vs Other Quadrilaterals

A parallelogram is one of several types of four-sided figures. The table below shows how it compares with related shapes.

Common Confusions

A few misconceptions about parallelograms appear often:

• Is every rectangle a parallelogram? Yes. A rectangle satisfies all parallelogram conditions and adds the extra requirement that all angles are 90°. The reverse is not true — not every parallelogram is a rectangle.

• Is a trapezium a parallelogram? No. A trapezium has only one pair of parallel sides; a parallelogram requires two.

• Does Area = ½ × d₁ × d₂ work for any parallelogram? No. That formula applies only when the diagonals are perpendicular — true for a rhombus and a square. For a general parallelogram, the correct formula includes sin(α): Area = ½ × d₁ × d₂ × sin(α).

• Are the diagonals of a parallelogram equal in length? Not in general. Diagonals are equal only in rectangles and squares.

When You'll See Parallelograms in the Curriculum

Parallelograms are introduced in Grade 3 as one category of quadrilateral (CCSS 3.G.A.1) and revisited in Grade 4 for classification by parallel sides and angles (CCSS 4.G.A.2).

In the Indian curriculum, parallelograms appear formally in Class 8 (NCERT Chapter 3 - Understanding Quadrilaterals) and again in Class 9 (NCERT Chapter 8 — Quadrilaterals), where their properties are proved formally. They also appear in higher mathematics through coordinate geometry, vector addition (the parallelogram law), and linear algebra.