Quadrilaterals Guide | Types, Properties & Formulas

A quadrilateral is a polygon with four sides and four angles. It is one of the most fundamental shapes in geometry and comes in various types, each with its unique properties. Whether you’re studying geometry for academic purposes or simply curious about the different quadrilaterals, this guide covers the essential types and properties.

What is a Quadrilateral?

A quadrilateral is any closed shape that has four sides. The sum of the interior angles of any quadrilateral is always 360°. Some of the most common types of quadrilaterals include the square, rectangle, parallelogram, rhombus, and trapezium.

Types of Quadrilaterals

Quadrilaterals come in different shapes, each with distinct characteristics. Let’s explore the key types:

1. Square: The Regular Quadrilateral

A square is a regular quadrilateral where all four sides are of equal length, and each interior angle measures 90°.

Key Properties of a Square:

Sides: All sides are equal.
Angles: All angles are 90°.
Diagonals: The diagonals bisect each other at right angles (perpendicular) and are of equal length.
Perimeter: $Perimeter=4a\text{Perimeter} = 4a$
Area: $Area=a2\text{Area} = a^2$
Diagonal Length: $Diagonal=2a\text{Diagonal} = \sqrt{2}a$

2. Parallelogram: Opposite Sides Equal and Parallel

A parallelogram is a quadrilateral in which opposite sides are equal and parallel.

Key Properties of a Parallelogram:

Opposite sides: Equal and parallel.
Opposite angles: Equal.
Diagonals: The diagonals bisect each other but do not necessarily have the same length.
Area: $Area=Base×Height\text{Area} = \text{Base} \times \text{Height}$
Perimeter: $Perimeter=2(Side1+Side2)\text{Perimeter} = 2(\text{Side}_1 + \text{Side}_2)$
Diagonals Relation: $AC^2 + BD^2 = AB^2 + BC^2 + CD^2 + DA^2$

3. Rectangle: A Special Parallelogram

A rectangle is a type of parallelogram where all four angles are 90°. It has the same properties as a parallelogram but with additional symmetry.

Key Properties of a Rectangle:

Angles: All angles are 90°.
Sides: Opposite sides are equal.
Diagonals: The diagonals are equal in length but do not necessarily bisect each other at right angles.
Area: $Area=Length×Breadth\text{Area} = \text{Length} \times \text{Breadth}$
Perimeter: $Perimeter=2(Length+Breadth)\text{Perimeter} = 2(\text{Length} + \text{Breadth})$

4. Rhombus: Equal Sides, Opposite Angles Equal

A rhombus is a parallelogram in which all sides are of equal length. The diagonals are perpendicular and divide the rhombus into four right-angled triangles.

Key Properties of a Rhombus:

Sides: All sides are equal.
Angles: Opposite angles are equal.
Diagonals: Diagonals are perpendicular and divide the rhombus into four triangles.
Area: $Area=12×Diagonal1×Diagonal2\text{Area} = \frac{1}{2} \times \text{Diagonal}_1 \times \text{Diagonal}_2$

5. Trapezium: One Pair of Parallel Sides

A trapezium (also called a trapezoid in some regions) is a quadrilateral with one pair of opposite sides parallel.

Key Properties of a Trapezium:

Parallel sides: The parallel sides are called bases.
Lateral sides: The non-parallel sides are called lateral sides.
Height: The perpendicular distance between the bases is called the height.
Midline: The midline (EF), joining the midpoints of the lateral sides, is equal to half the sum of the lengths of the bases.
Isosceles Trapezium: If the non-parallel sides are equal, the trapezium is called an isosceles trapezium.

Solved Examples: Practice Makes Perfect

Here are some practice problems that will help you understand how to apply the properties of quadrilaterals:

Q1: Diagonals Bisect at Right Angles

If the diagonals of a quadrilateral bisect each other at right angles, name the quadrilateral.

Solution: The quadrilateral is a rhombus.

Q2: Find the Measure of an Angle in a Quadrilateral

In quadrilateral PQRS, if $∠P=60∘\angle P = 60^\circ$ and the angles $∠Q:∠R:∠S=2:3:7\angle Q : \angle R : \angle S = 2:3:7$ , find the measure of $∠S\angle S$ .

Solution:
The total sum of the angles of a quadrilateral is 360°. Solve for $∠S\angle S$ .

Conclusion: Mastering Quadrilaterals for Geometry Success

Quadrilaterals are fundamental shapes in geometry with various properties and applications. From squares to trapeziums, each type of quadrilateral has unique characteristics that make it important in both academic and real-world contexts. By mastering the properties, formulas, and applications of these quadrilaterals, you will be well-equipped to tackle geometry problems with confidence.