A parallelogram is a quadrilateral (a polygon with four sides) in which each pair of opposite sides is both parallel and equal in length. This specific pairing gives the shape its name and its geometric properties. Parallelograms appear frequently in mathematics, engineering, computer graphics, and physics, forming the foundation for topics such as vector addition and transformations.
Special cases of parallelograms include rectangles, rhombuses, and squares, which meet additional criteria such as right angles or equal side lengths.
This guide covers definitions, types, properties, how to calculate area and perimeter, how to find diagonals and height, and includes real-world examples, misconceptions, coordinate geometry strategies, and helpful comparison charts.
Key Takeaways: Parallelogram
- A parallelogram is a quadrilateral in which each pair of opposite sides is both parallel and equal in length.
- Opposite angles are equal; adjacent angles are supplementary (sum to 180°).
- Diagonals bisect each other, though they are not necessarily equal or perpendicular.
- Area is base × height; perimeter is 2 × (base + side).
- Special types include the rectangle, rhombus, and square.
- Coordinate geometry tools (slope, distance, midpoint) help verify whether a figure is a parallelogram.
- Parallelograms are seen in real-world structures, physics problems, and graphic design.
What Is a Parallelogram?
A parallelogram is a four-sided polygon (quadrilateral) in which both pairs of opposite sides are parallel and equal in length. This configuration results in symmetry and leads to other important properties:
- Opposite angles are equal.
- Consecutive angles are supplementary.
- Diagonals bisect each other (they intersect at their midpoints).
The term comes from Greek parallēlogrammon, meaning “bounded by parallel lines.”
How to Identify a Parallelogram
A quadrilateral is a parallelogram if any one of the following is true:
- Both pairs of opposite sides are parallel.
- Both pairs of opposite sides are equal in length.
- One pair of opposite sides is both equal and parallel.
- Diagonals bisect each other.
- Opposite angles are equal.
- Consecutive angles are supplementary.
These identification rules are used in both synthetic and coordinate geometry.
Examples of Shapes That Are and Are Not Parallelograms
Shapes That ARE Parallelograms
| Shape | Why It Qualifies |
|---|---|
| Square | All sides equal, angles 90°, opposite sides parallel. |
| Rectangle | Opposite sides equal and parallel; all angles 90°. |
| Rhombus | All sides equal; opposite sides parallel; diagonals perpendicular. |
| Generic Parallelogram | Opposite sides and angles equal; no requirement for right angles or equal diagonals. |
Shapes That Are NOT Parallelograms
| Shape | Why It Does Not Qualify |
|---|---|
| Trapezoid | Only one pair of opposite sides is parallel. |
| Kite | Adjacent sides equal; no parallel pairs. |
| Regular Hexagon | Has three pairs of parallel sides, but is a six-sided polygon and not a quadrilateral. |
| Irregular Quadrilateral | No side or angle symmetry; opposite sides not parallel. |
Types of Parallelograms
Parallelograms can take several specialized forms:
1. Rectangle
- Opposite sides equal and parallel.
- Four 90° angles.
- Diagonals equal and bisect each other.
2. Rhombus
- Four equal sides.
- Opposite angles equal.
- Diagonals bisect each other at 90°, but not equal.
3. Square
- All properties of a rectangle and a rhombus.
- All sides equal and all angles 90°.
- Diagonals equal and perpendicular.
4. General Parallelogram
- Opposite sides and angles equal.
- Diagonals bisect each other.
- Angles not necessarily right; diagonals not necessarily equal.
Properties of a Parallelogram
| Property | Description |
|---|---|
| Opposite sides | Equal and parallel |
| Opposite angles | Equal |
| Adjacent angles | Supplementary (sum to 180°) |
| Diagonals | Bisect each other |
| Interior angle sum | 360° |
| Rotational symmetry | 180° |
| Line symmetry | Only if it’s a rectangle, square, or rhombus |
How to Calculate the Area of a Parallelogram
Area measures how much space a parallelogram covers. It is not enough to know side lengths. You must also know the height, which is the perpendicular distance from a base to its opposite side.
Area Formula
Area = base × height
How to Find the Height
When Area and Base Are Known:
Height = Area / Base
When Two Sides and Included Angle Are Known:
If you know two adjacent sides and the angle θ between them:
Area = ab sin(θ)
Height=a sin(θ)
Examples
Example 1:
Base = 10 cm, Height = 4 cm
Area = 10 × 4 = 40 cm2
Example 2 (Using Angle):
Sides = 6 cm and 8 cm, angle = 60°
Area = 6 × 8 × sin(60°) = 48 × 0.866 ≈ 41.57 cm2
How to Find the Perimeter of a Parallelogram
The perimeter is the total length around the parallelogram.
Formula
Perimeter = 2(a + b)
Where a and b are the lengths of adjacent sides.
Example:
If a = 7 cm, b = 5 cm:
Perimeter = 2(7 + 5) = 24 cm
How to Find the Height of a Parallelogram
Height is the vertical distance from the base to the opposite side. It is crucial in calculating the area.
Height Formulas
- From area and base:
h = Area / Base - From angle and side:
h = as in(θ)
How to Find the Diagonals of a Parallelogram
Diagonals cross from one vertex to the opposite. They bisect each other but are only equal in rectangles and squares.
Coordinate Geometry Method
If diagonals connect points A(x1,y1) and C(x3,y3), use:
Diagonal length = [(x3 − x1)2 + (y3 − y1)2]1/2
Do this for both diagonals AC and BD.
Parallelograms in Coordinate Geometry
Coordinate geometry allows you to prove a shape is a parallelogram using algebra.
Tools
- Slope Formula (for parallelism):
m = (y2 − y1) / (x2 − x1) - Distance Formula (for side lengths):
d = [(x2 − x1)2 + (y2 − y1)2]1/2 - Midpoint Formula (to check diagonal bisection):
M = ((x1 + x2) / 2, (y1 + y2) / 2)
Example
Given vertices: A(0, 0), B(4, 0), C(5, 3), D(1, 3)
- AB ∥ CD (same slope)
- BC ∥ DA
- Diagonals intersect at same midpoint
⇒ This is a parallelogram.
Real-World Examples and Applications
| Field | Example |
|---|---|
| Engineering | Parallelogram linkages in machinery |
| Physics | Vector addition using parallelograms |
| Architecture | Slanted supports, framing structures |
| Crystallography | Lattice unit cells in 2D layers |
| Design/Art | Tiling and tessellation patterns |
Comparison Table: Quadrilaterals
| Property | Generic Parallelogram | Rectangle | Rhombus | Square | Trapezoid |
|---|---|---|---|---|---|
| Opposite sides parallel | Yes | Yes | Yes | Yes | One pair |
| All angles 90° | No | Yes | No | Yes | Not required |
| All sides equal | No | No | Yes | Yes | Not required |
| Diagonals equal | No | Yes | No | Yes | Not required |
| Diagonals perpendicular | No | No | Yes | Yes | No |
| Diagonals bisect each other | Yes | Yes | Yes | Yes | No |
Common Misconceptions
| Misconception | Correction |
|---|---|
| All parallelograms have right angles | Only rectangles and squares do. |
| All parallelograms are rectangles | Rectangles are a subtype; not all parallelograms have 90° angles. |
| Height is the side of the figure | Height is the perpendicular distance, not necessarily a side. |
| A hexagon cannot have parallel sides | A regular hexagon has three pairs of parallel sides, but it is not a parallelogram because it has six sides. |
| If opposite sides are equal, it’s a parallelogram | They must also be parallel. |
Frequently Asked Questions (FAQs)
Q1. Is a square a parallelogram?
Yes, with all sides and angles equal.
Q2. Can a parallelogram have right angles?
Yes, if it is a rectangle or square.
Q3. Are diagonals always equal?
No, only in rectangles and squares.
Q4. Can a parallelogram be concave?
No. All internal angles of a parallelogram are less than 180°.
Q5. Is a hexagon a parallelogram?
No, it has six sides. Parallelograms are quadrilaterals.
Q6. Is every rhombus a parallelogram?
Yes. All rhombuses have opposite sides parallel.
Q7. Is height the same as side length?
Not necessarily. Height is measured perpendicularly to the base.
References
- Byer, Owen; Lazebnik, Felix; Smeltzer, Deirdre (2010). Methods for Euclidean Geometry. Mathematical Association of America.
- Chen, Zhibo; Liang, Tian (2006). “The converse of Viviani’s theorem“. The College Mathematics Journal. 37(5): 390–391.
- Usiskin, Zalman; Griffin, Jennifer (2008). The Classification of Quadrilaterals. A Study of Definition. Information Age Publishing.