Quadrilaterals

A quadrilateral is any polygon with four sides (edges) and four vertices (corners). The word “quad” means four, and “lateral” means side.

Imagine a flexible wire bent into four segments — it becomes a quadrilateral as long as the ends connect and don’t cross.

Every quadrilateral has:

• 4 sides
• 4 angles
• 2 diagonals
• Angle sum of 360°

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Classification of Quadrilaterals

Quadrilaterals can be broadly classified based on side lengths, angles, and symmetry.

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Angle Properties

• Sum of interior angles of any quadrilateral =

  $$360^\circ$$

• Sum of exterior angles (taken one at each vertex) =

  $$360^\circ$$

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Key Properties by Type

Parallelogram

• Opposite sides and angles equal
• Diagonals bisect each other
• Each diagonal divides it into two congruent triangles

Rectangle

• All properties of parallelogram
• All angles = 90°
• Diagonals are equal in length

Rhombus

• All sides equal
• Opposite angles equal
• Diagonals bisect each other perpendicularly

Square

• All sides and angles equal (90°)
• Diagonals are equal, bisect perpendicularly, and divide square into 4 right isosceles triangles

Trapezium

• Only one pair of opposite sides is parallel
• Non-parallel sides are legs, parallel sides are bases

Kite

• One diagonal is the axis of symmetry
• Diagonals intersect at right angles
• One pair of opposite angles is equal

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Key Formulas

Area of a Parallelogram

$$\text{Area} = \text{Base} \times \text{Height}$$

Area of a Rectangle

$$\text{Area} = \text{Length} \times \text{Breadth}$$

Area of a Rhombus

$$\text{Area} = \frac{1}{2} \times d_1 \times d_2 \quad \text{(where } d_1 \text{ and } d_2 \text{ are diagonals)}$$

Area of a Square

$$\text{Area} = a^2, \quad \text{where } a = \text{side}$$ $$\text{Diagonal} = a\sqrt{2}$$

Area of a Trapezium

$$\text{Area} = \frac{1}{2} \times (\text{Sum of parallel sides}) \times \text{Height}$$

Area of a Kite

$$\text{Area} = \frac{1}{2} \times d_1 \times d_2$$

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Diagonals

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Visual Understanding

Square Example:

A ------- B
|         |
|         |
D ------- C

• All sides =
• All angles = 90°
• Diagonals $AC$ and $BD$ bisect each other at 90°

Trapezium Example:

A -------------- B
 \            /
  \          /
   D--------C

• $AB \parallel DC$, but $AD \not\parallel BC$

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Conceptual Tips & Common Mistakes

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Shortcuts & Tricks

• In coordinate geometry:

  • Use slope for parallelism
  • Use distance formula for diagonals
  • Use midpoint formula to test if diagonals bisect each other

• Diagonals bisecting + perpendicular = likely rhombus or square

• If all sides equal and diagonals equal → Square
  If only diagonals equal → Rectangle

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Examples

Example 1:

Find area of a square with side 10 cm.

$$A = a^2 = 10^2 = 100 \text{ cm}^2$$

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Example 2:

In a rhombus, diagonals are 12 cm and 16 cm. Find the area.

$$A = \frac{1}{2} \times 12 \times 16 = 96 \text{ cm}^2$$

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Example 3:

A trapezium has parallel sides 10 cm and 6 cm, and height 5 cm. Find area.

$$A = \frac{1}{2} \times (10 + 6) \times 5 = \frac{1}{2} \times 16 \times 5 = 40 \text{ cm}^2$$

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Example 4:

Which quadrilateral has all sides equal and diagonals perpendicular but not necessarily equal?

• Answer: Rhombus