Quadrilateral

Introduction to Quadrilateral

A quadrilateral is a special type of polygon that has four sides, four corners, and four angles. The word 'quadrilateral' comes from the Latin words 'quadri', meaning 'four', and 'latus', meaning 'side'. It is one of the most common shapes we see in real life with geometry.

An important feature of a quadrilateral is that the sum of all its interior angles is always 360°. A quadrilateral can have sides and angles that can be equal (like a square) or different (like a trapezium). Depending on these properties, the quadrilateral is divided into different types, such as square, rectangle, rhombus, parallelogram, kite, and trapezium.

We can observe the quadrilateral around us in our daily lives. Objects such as windows, books, walls, tiles, and even the road are shaped like quadrilaterals. In this article, we will learn about types of quadrilaterals, their properties, and examples to make the concept easy and clear.

Table of Contents

• Definition
• Types of Quadrilaterals
• Properties of Quadrilaterals
• Sides and Angles of Quadrilaterals
• Quadrilateral Formulas
• Solved Examples
• Practice Questions
• FAQs on Quadrilaterals

 

Definition

A quadrilateral is a flat (plane) figure with four sides and four corners (vertices). The angles of a quadrilateral are present in these four corners. For example, if we take a quadrilateral ABCD, the angles are ∠A, ∠B, ∠C, and ∠D. The four sides are AB, BC, CD, and DA.

If we go along with the opposite corners (vertices) of a quadrilateral, we get diagonals. In quadrilateral ABCD, there are diagonals AC and BD.

A quadrilateral can be of different types. There are some regular shapes, such as a square, a rectangle, a rhombus, a parallelogram, a kite, and a trapezium. Other irregular shapes may occur that do not have equal sides or equal angles.

 

Types of Quadrilaterals

The quadrilaterals are divided into different types based on the length of the sides and the measure of their angles. Since "quad" means four, all quadrilaterals have four sides, and the sum of all angles is always 360°.

The main types of quadrilaterals are:

• Trapezium

• Parallelogram

• Square 

• Rectangle

• Rhombus

• Kite
  
  

Convex, Concave, and Intersecting Quadrilaterals

The quadrilaterals can also be grouped differently:

• Convex Quadrilateral: Both diagonals lie completely inside the shape. Example: square, rectangle.

• Concave Quadrilateral: At least one diagonal lies partly or completely within the shape. Example: A dart-shaped figure.

• Intersecting Quadrilateral: In this way, two non-adjacent sides cross each other. They are also called self-intersecting or crossed quadrilaterals. Example: A bow-tie shape.

 

Properties of Quadrilaterals

Let's understand the properties of a quadrilateral with an example:

• It has four pages: AB, BC, CD, and DA

• It has four vertices (corners): A, B, C, and D

• It has four angles: ∠ABC, ∠BCD, ∠CDA, and ∠DAB

• ∠A and ∠B are called adjacent angles

• ∠A and ∠C are called opposite angles

• AB and CD are opposite sides

• AB and BC are adjacent sides

In short, a quadrilateral is a 4-sided closed figure.

• Each quadrilateral has 4 sides, 4 vertices, and 4 angles

• The sum of all the interior angles of a quadrilateral is always 360°.

Properties of a square

• All four sides are equal in length.

• The opposite sides are parallel to each other.

• All four angles are 90° (right angles).

• Diagonals are equal and cut each other at right angles.

Properties of a Rectangle 

• The opposite side is equal in length

• The opposite sides are parallel

• All four angles are 90°.

• Diagonals are equal and cut each other into two halves

Properties of a Rhombus

• All four sides are equal in length. 

• The opposite sides are parallel

• The opposite angles are equal. 

• The sum of two adjacent angles is 180°.

• Diagonals cut each other at right angles.

Properties of a Parallelogram

• The opposite side is equal in length

• The opposite sides are parallel

• The opposite angles are equal. 

• The sum of two adjacent angles is 180°.

• Diagonals cut each other into two halves

Properties of a Trapezium 

• Only one pair of opposite sides is parallel

• The sum of two adjacent angles on the same side is 180°.

• Whether the diagonals may be equal or not, they cut each other at a ratio.

Properties of a Kite 

• It has two pairs of equal adjacent sides.

• One  diagonal is longer and cuts the shorter diagonal into 2  equal parts

• Only one pair of opposite angles is equal.
  
  

Sides and Angles of Quadrilaterals

Each quadrilateral has 4 sides, 4 corners (vertices), and 4 angles. But their sides and angles may not always be the same. Here are some important points:

• Square: All 4 sides are equal, all 4 angles are 90°, and both diagonals bisect each other.

• Rectangles: The opposite sides are equal and parallel, all angles are 90°, and diagonals bisect each other.

• Rhombus: All 4 sides are equal, the opposite angles are equal, and the diagonals bisect each other at a right angle.

• Parallelograms: The opposite sides are equal and parallel, the opposite angles are equal, and the diagonals bisect each other.

• Trapezium: Only one pair of opposite sides is parallel, and the other sides are not equal.

 

Quadrilateral Formulas

There are two important measurements of a quadrilateral: area and perimeter.

1. Area of Quadrilateral

• Square → Side × Side 

• Rectangle → length × breadth 

• Parallelogram → base × height

• Rhombus → (1/2) × diagonal 1 × diagonal 2

• Kite → (1/2) × diagonal 1 × diagonal 2

• Trapezium → (1/2) × (sum of parallel sides) × height

2. Perimeter of a Quadrilateral

• Class → 4 × side 

• Rectangle → 2 × (length + breadth)

• Parallelogram → 2 × (base + side)

• Rhombus → 4 × side

• Kite → 2 × (a + b), where a and b are pairs of equal sides

• Trapezium → Sum of all 4 pages

 

Solved Examples

Example 1: Find the perimeter of a square whose side is 6 cm.

Solution: Perimeter of a square = 4 × side

• = 4 × 6 

• = 24 cm

• So, the perimeter of the square is 24 cm.

Example 2: The length of a rectangle is 12 cm and the breadth is 8 cm. Find its area.

Solution: Area of a rectangle = length × breadth 

• = 12 × 8 

• = 96 cm²

• So, the area of the rectangle is 96 cm².

Example 3: The diagonals of a rhombus are 10 cm and 8 cm. Find its area.

Solution: Area of a rhombus = (1/2) × diagonal 1 × diagonal 2

• = (1/2) × 10 × 8

• =  40 cm²

• So, the area of the rhombus is 40 cm²

 

Practice Questions

1. Find the perimeter of a rectangle with a length of 15 cm and a breadth of 9 cm.

2. The side of a square is 7 cm. Find out its area.

3. A trapezium has parallel sides of 12 cm and 8 cm, and the height is 5 cm. Find out its area.

4. Find the perimeter of a rhombus if each side measures 9 cm. 

5. The diagonals of a kite are 6 cm and 4 cm. Find out its area.

 

Frequently Asked Questions on Quadrilaterals

1. What are the 7 types of quadrilaterals?

Answer: The 7 types of quadrilaterals are:

• Square

• Rectangle

• Rhombus

• Parallelogram

• Trapezium (Trapezoid)

• Kite

• General Quadrilateral

 

2. What is called a quadrilateral?

Answer: A quadrilateral is a closed shape in geometry that has four sides, four corners (vertices), and four angles.

 

3. How do you identify a quadrilateral?

Answer: To identify a quadrilateral, check if the shape has:

• Exactly 4 sides

• 4 angles

• 4 corners (vertices)

If these are present, then the shape is a quadrilateral.

 

4. Is a quadrilateral 180 or 360 degrees?

Answer: The sum of all angles in a quadrilateral is always 360 degrees.

 

5. What is a 180- to 360-degree angle called?

Answer: An angle that is more than 180 degrees but less than 360 degrees is called a reflex angle.