Quadrilaterals

For Elementary Students

What is a Quadrilateral?

A quadrilateral is any shape with 4 straight sides and 4 corners (vertices).

Quad means "four" — just like quadruplets are four babies!

All of these are quadrilaterals:

• Squares
• Rectangles
• Trapezoids
• Diamonds (rhombuses)
• And more!

Basic Properties

Every quadrilateral has:

• 4 sides
• 4 corners (vertices)
• 4 angles
• Angles add up to 360° total

Square

A square has four equal sides and four right angles (90° corners).

Properties:

• All 4 sides are the same length
• All 4 angles are 90° (right angles)
• Opposite sides are parallel

Examples in real life:

• Checkerboard square
• Square tile
• Post-it note

Rectangle

A rectangle has four right angles, with opposite sides equal.

Properties:

• Opposite sides are equal length
• All 4 angles are 90°
• Longer than it is wide (or vice versa)
• Opposite sides are parallel

Examples in real life:

• Door
• Book
• Smartphone screen
• Dollar bill

Note: A square is a special type of rectangle (where all sides happen to be equal)!

Trapezoid

A trapezoid has one pair of parallel sides.

Properties:

• At least one pair of parallel sides
• The parallel sides can be different lengths
• The other two sides are usually slanted

Examples in real life:

• Some table tops
• Bucket shape (viewed from side)

Rhombus (Diamond)

A rhombus has four equal sides but not right angles (usually).

Properties:

• All 4 sides are equal
• Opposite sides are parallel
• Opposite angles are equal
• Looks like a pushed-over square

Examples in real life:

• Diamond shape ♦
• Some kites

Parallelogram

A parallelogram has opposite sides that are parallel and equal.

Properties:

• Opposite sides are parallel
• Opposite sides are equal length
• Opposite angles are equal
• Looks like a leaning rectangle

Examples in real life:

• Slanted bookshelf
• Some road signs

For Junior High Students

Quadrilateral Family Tree

All quadrilaterals fit into a hierarchy:

Quadrilateral (any 4-sided shape)
├── Trapezoid (1 pair parallel sides)
└── Parallelogram (2 pairs parallel sides)
    ├── Rectangle (parallelogram with right angles)
    │   └── Square (rectangle with equal sides)
    └── Rhombus (parallelogram with equal sides)
        └── Square (rhombus with right angles)

Notice: A square is both a rectangle AND a rhombus!

Detailed Properties Table

Diagonals

The diagonals are lines connecting opposite corners.

Square:

• Diagonals are equal length
• Diagonals bisect each other (cut in half)
• Diagonals are perpendicular (meet at 90°)

Rectangle:

• Diagonals are equal length
• Diagonals bisect each other
• Diagonals are NOT perpendicular

Rhombus:

• Diagonals bisect each other
• Diagonals are perpendicular
• Diagonals are NOT equal

Parallelogram:

• Diagonals bisect each other
• Diagonals are NOT equal
• Diagonals are NOT perpendicular

Special Types of Trapezoids

Isosceles trapezoid:

• The non-parallel sides are equal length
• Base angles are equal
• Diagonals are equal

Right trapezoid:

• Has two right angles

Perimeter of Quadrilaterals

Perimeter = sum of all side lengths

Square: P = 4 × side

• Example: side = 5 → P = 4 × 5 = 20

Rectangle: P = 2 × length + 2 × width

• Example: length = 8, width = 3 → P = 2(8) + 2(3) = 16 + 6 = 22

Other quadrilaterals: Add all four sides

• Example: sides are 3, 5, 4, 6 → P = 3 + 5 + 4 + 6 = 18

Area of Quadrilaterals

Square: A = side²

• Example: side = 4 → A = 4² = 16

Rectangle: A = length × width

• Example: 6 × 3 → A = 18

Parallelogram: A = base × height

• (Height is perpendicular distance between parallel sides)

Trapezoid: A = ½ × (base₁ + base₂) × height

We will learn more about area formulas later.

Identifying Quadrilaterals

Strategy: Check properties in order:

1. How many parallel sides?

   • None → irregular quadrilateral
   • 1 pair → trapezoid
   • 2 pairs → parallelogram (or special type)

2. Are all angles 90°?

   • Yes → rectangle (or square)
   • No → rhombus or parallelogram

3. Are all sides equal?

   • Yes → square or rhombus
   • No → rectangle or parallelogram

Real-World Applications

Architecture: Buildings use rectangles and squares for stability

Art: Quadrilaterals in patterns and designs

Sports: Baseball diamond (actually a square!), basketball court (rectangles)

Everyday: Screens, papers, tables, windows

Practice