Class 8 - Quadrilateral

A quadrilateral is a shape with four sides and four corners like a square, rectangle, or diamond. You can see them on flags, boxes, and tabletops all the time. Some are straight and neat while others are wobbly, but they all have features like angles that add up to 360 degrees. A polygon with four line segments is called a quadrilateral.

Table of Contents

• What is a Quadrilateral?
• Types of Quadrilateral
• Properties of Quadrilaterals
• Sides and Angles of Quadrilaterals
• Area of a Quadrilateral
• Quadrilateral Solved Examples
• Practice Questions on Quadrilateral

What is a Quadrilateral?

A quadrilateral is a four sided polygon with four angles and four vertices. When naming it, always go in order around the shape. Take quadrilateral ABCD: you can call it ABCD, BCDA, ADCB, or DCBA. These trace the vertices clockwise or counterclockwise without skipping.

But skip names like ACBD or DBA they wont follow the sequence and don't follow the edge order.

In ABCD, the sides are AB, BC, CD, and DA. It also has two diagonals: AC (connecting A to C) and BD (B to D).

Read more: Important Questions on Quadrilaterals - Class 8

Types of Quadrilaterals

Quadrilaterals are classified into different types based on their sides and angles. Here are the main types:

1. Square

A square is a quadrilateral with four equal sides and four right angles (90 degrees each). All sides are parallel to adjacent sides, and the diagonals are equal and bisect each other at right angles.

Properties:

• All four sides are equal
• All four angles are 90°
• Diagonals are equal and bisect each other
• Diagonals bisect the angles
• The shape has four lines of symmetry

2. Rectangle

A rectangle is a quadrilateral with four right angles. Opposite sides are equal and parallel. The diagonals are equal and bisect each other, but they do not meet at right angles.

Properties:

• Opposite sides are equal and parallel
• All four angles are 90°
• Diagonals are equal in length
• Diagonals bisect each other
• It has two lines of symmetry

3. Parallelogram

A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. Opposite angles are also equal, but the angles are not 90 degrees.

Properties:

• Opposite sides are parallel and equal
• Opposite angles are equal
• Consecutive angles are supplementary (add up to 180°)
• Diagonals bisect each other
• It does not have lines of symmetry (except special cases)

4. Rhombus

A rhombus is a parallelogram with all four sides equal in length. The angles are not necessarily 90 degrees. The diagonals bisect each other at right angles and bisect the angles of the rhombus.

Properties:

• All four sides are equal
• Opposite sides are parallel
• Opposite angles are equal
• Diagonals bisect each other at right angles
• Diagonals bisect the vertex angles
• It has two lines of symmetry

5. Trapezium (Trapezoid)

A trapezium is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the non-parallel sides are called legs. Trapeziums are common in engineering and architecture.

Properties:

• One pair of parallel sides
• The sum of angles on the same leg is 180°
• It has one line of symmetry only in isosceles trapezium
• Diagonals may or may not be equal

6. Kite

A kite is a quadrilateral with two pairs of consecutive sides that are equal in length. One diagonal (the axis of symmetry) bisects the other diagonal at right angles.

Properties:

• Two pairs of consecutive sides are equal
• One diagonal is the axis of symmetry
• One diagonal bisects the other at right angles
• One pair of opposite angles are equal
• It has one line of symmetry

Properties of Quadrilaterals

Every quadrilateral shares certain fundamental properties:

1. Sum of Interior Angles

The sum of all four interior angles in any quadrilateral is always 360 degrees. This property holds true for all types of quadrilaterals, whether regular or irregular.

Formula: ∠A + ∠B + ∠C + ∠D = 360°

2. Diagonals

Every quadrilateral has exactly two diagonals. These are line segments that connect opposite vertices. The properties of diagonals vary depending on the type of quadrilateral.

3. Perimeter

The perimeter of a quadrilateral is the sum of all four sides. It represents the total distance around the shape.

Formula: Perimeter = Side1 + Side2 + Side3 + Side4

4. Exterior Angles

The sum of exterior angles of any quadrilateral is 360 degrees, just like any other polygon.

Special Properties by Type

Sides and Angles of Quadrilaterals

Understanding Sides

The sides of a quadrilateral are the four line segments that form the shape. They connect the four vertices. Different types of quadrilaterals have different relationships between their sides.

Examples:

• Square: All four sides are equal (AB = BC = CD = DA)
• Rectangle: Opposite sides are equal (AB = CD, BC = DA)
• Rhombus: All four sides are equal (AB = BC = CD = DA)
• Parallelogram: Opposite sides are equal (AB = CD, BC = DA)
• Trapezium: Only one pair of sides is parallel
• Kite: Two pairs of consecutive sides are equal

Understanding Angles

The angles of a quadrilateral are the four interior angles at each vertex. The total of all angles is always 360 degrees.

Examples:

• Square: All angles are 90° each
• Rectangle: All angles are 90° each
• Parallelogram: Opposite angles are equal
• Rhombus: Opposite angles are equal (may not be 90°)
• Trapezium: Angles vary depending on the shape
• Kite: One pair of opposite angles is equal

Angle Relationships

In many quadrilaterals, there are special angle relationships:

1. Consecutive Angles in Parallelogram: Are supplementary (add up to 180°)
2. Opposite Angles in Parallelogram/Rhombus: Are equal
3. Angles in Square/Rectangle: All are right angles (90°)

Area of a Quadrilateral

The area of a quadrilateral is the amount of space enclosed within the shape. Different quadrilaterals have different formulas for calculating area.

General Formula Using Diagonals

For any quadrilateral when diagonals and the angle between them are known:

Formula: Area = (d₁ × d₂ × sin(θ)) / 2

Where d₁ and d₂ are the diagonals and θ is the angle between them.

Quadrilateral Solved Examples

Example 1: Finding Area of a Square

Problem: Find the area of a square with side length 7 cm.

Solution:

• Given: Side = 7 cm
• Formula: Area = side²
• Area = 7 × 7 = 49 cm²
• Answer: The area of the square is 49 cm²

Example 2: Finding Angles in a Quadrilateral

Problem: In a quadrilateral ABCD, angles A, B, and C are 80°, 95°, and 110° respectively. Find angle D.

Solution:

• Given: ∠A = 80°, ∠B = 95°, ∠C = 110°
• We know: ∠A + ∠B + ∠C + ∠D = 360°
• 80° + 95° + 110° + ∠D = 360°
• 285° + ∠D = 360°
• ∠D = 360° - 285° = 75°
• Answer: Angle D is 75°

Example 3: Finding Area of a Rectangle

Problem: A rectangular garden has length 12 m and width 8 m. Find the area.

Solution:

• Given: Length = 12 m, Width = 8 m
• Formula: Area = length × width
• Area = 12 × 8 = 96 m²
• Answer: The area of the garden is 96 m²

Example 4: Finding Area of a Trapezium

Problem: A trapezium has parallel sides of 5 cm and 9 cm, with height 4 cm. Calculate the area.

Solution:

• Given: Parallel side a = 5 cm, Parallel side b = 9 cm, Height h = 4 cm
• Formula: Area = ((a + b) / 2) × h
• Area = ((5 + 9) / 2) × 4
• Area = (14 / 2) × 4
• Area = 7 × 4 = 28 cm²
• Answer: The area of the trapezium is 28 cm²

Example 5: Finding Area of a Rhombus Using Diagonals

Problem: A rhombus has diagonals measuring 10 cm and 6 cm. Find the area.

Solution:

• Given: Diagonal d₁ = 10 cm, Diagonal d₂ = 6 cm
• Formula: Area = (d₁ × d₂) / 2
• Area = (10 × 6) / 2
• Area = 60 / 2 = 30 cm²
• Answer: The area of the rhombus is 30 cm²

Example 6: Finding Area of a Parallelogram

Problem: A parallelogram has base 15 cm and height 8 cm. Find the area.

Solution:

• Given: Base = 15 cm, Height = 8 cm
• Formula: Area = base × height
• Area = 15 × 8 = 120 cm²
• Answer: The area of the parallelogram is 120 cm²

Practice Questions on Quadrilateral

1: What is the sum of interior angles of a quadrilateral?

2: Name the quadrilateral with all four sides equal and all angles 90°.

3: A quadrilateral has angles 75°, 85°, 100°, and x°. Find the value of x.

4: Which quadrilateral has exactly one pair of parallel sides?

5: Find the area of a square with side 9 cm.

Conclusion

Quadrilaterals are essential shapes in geometry and mathematics. Understanding their properties, types, and formulas helps solve real-world problems in construction, engineering, and design. Whether you are calculating the area of a room, designing a garden, or solving geometry problems, knowledge of quadrilaterals is invaluable. By practicing the solved examples and working through the practice questions, you can master the concepts of quadrilaterals and apply them confidently in various situations.