Quadrilaterals and Polygons

Quadrilaterals (four-sided figures) and polygons show up constantly in aptitude exams. The secret here is understanding the hierarchy — a square is a special rectangle, which is a special parallelogram. Once we see these relationships, properties cascade down and we don’t need to memorize everything separately.

The Quadrilateral Family Tree

The key insight: as we go down the tree, each shape has all the properties of its parent PLUS some extra ones. A square has every property of a rectangle AND every property of a rhombus.

Properties of Each Quadrilateral

Parallelogram

• Opposite sides are equal AND parallel
• Opposite angles are equal
• Adjacent angles are supplementary (add to 180°)
• Diagonals bisect each other (cut each other in half)
• Area = base × height

Rectangle (Parallelogram + all right angles)

• All angles = 90°
• Diagonals are equal and bisect each other
• Area = length × breadth
• Diagonal = √(l² + b²)

Rhombus (Parallelogram + all sides equal)

• All sides are equal
• Diagonals bisect each other at right angles (90°)
• Diagonals bisect the vertex angles
• Area = ½ × d₁ × d₂ (half the product of diagonals)

Square (Rectangle + Rhombus = everything)

• All sides equal, all angles = 90°
• Diagonals are equal, bisect at right angles, bisect vertex angles
• Area = side² = ½ × diagonal²
• Diagonal = side × √2

Trapezium (only one pair of parallel sides)

• One pair of parallel sides (called parallel sides or bases)
• Area = ½ × (sum of parallel sides) × height = ½(a + b) × h
• Isosceles trapezium: non-parallel sides are equal, diagonals are equal

Kite

• Two pairs of adjacent sides are equal
• One diagonal bisects the other at right angles
• Area = ½ × d₁ × d₂

Key Formulas

Diagonal Properties Quick Reference

Memory trick: As we go from parallelogram to square, diagonals gain more and more properties. Rectangle adds “equal,” rhombus adds “perpendicular,” and square has both.

Polygons

A polygon is any closed figure with straight sides. Quadrilaterals are 4-sided polygons. Let’s look at the general rules.

Angle Sum of a Polygon

Interior angle sum = (n - 2) × 180°, where n is the number of sides.

This makes sense: a triangle (3 sides) has 180°, a quadrilateral (4 sides) has 360°, a pentagon has 540°, and so on. We’re basically dividing the polygon into (n-2) triangles.

Regular Polygons

A regular polygon has all sides equal AND all angles equal.

• Each interior angle = (n - 2) × 180° / n
• Each exterior angle = 360° / n
• Number of diagonals = n(n - 3) / 2

Important fact: The sum of exterior angles of ANY convex polygon is always 360°, regardless of the number of sides.

Shortcut Methods and Tricks

Trick 1: Finding shape from diagonals If a question describes a quadrilateral by its diagonal properties, use the table above. “Diagonals bisect at right angles but aren’t equal” → Rhombus. “Diagonals are equal and bisect each other” → Rectangle.

Trick 2: Rhombus area shortcut If we know the diagonals of a rhombus, we can find the side using: side = ½√(d₁² + d₂²). This is because the diagonals form 4 right triangles with legs d₁/2 and d₂/2.

Trick 3: Finding exterior angle first For regular polygons, always find the exterior angle first (360°/n) — it’s simpler. Then interior = 180° - exterior.

Trick 4: Reverse problem — finding n from angle If given an interior angle and asked to find the polygon: exterior angle = 180° - interior angle, then n = 360° / exterior angle.

Worked Examples

Example 1: Parallelogram Area

A parallelogram has a base of 12 cm and a height of 8 cm. Find its area.

Area = base × height = 12 × 8 = 96 cm²

Note: the slant side is NOT the height! The height is the perpendicular distance between the parallel sides.

Example 2: Rhombus from Diagonals

The diagonals of a rhombus are 16 cm and 12 cm. Find the area and the side length.

Area = ½ × d₁ × d₂ = ½ × 16 × 12 = 96 cm²

Side = ½√(16² + 12²) = ½√(256 + 144) = ½√400 = ½ × 20 = 10 cm

(We used the 3-4-5 triplet here — half-diagonals are 8 and 6, so side = 10. Quick!)

Example 3: Trapezium Area

A trapezium has parallel sides of 10 cm and 16 cm, and the height between them is 9 cm. Find the area.

Area = ½ × (a + b) × h = ½ × (10 + 16) × 9 = ½ × 26 × 9 = 117 cm²

Example 4: Finding the Polygon

Each interior angle of a regular polygon is 140°. How many sides does it have?

Exterior angle = 180° - 140° = 40° Number of sides = 360° / 40° = 9 sides (nonagon)

Example 5: Number of Diagonals

How many diagonals does a decagon (10-sided polygon) have?

Diagonals = n(n - 3)/2 = 10 × 7 / 2 = 35 diagonals

Common Exam Variations

1. “Which quadrilateral…” — identifying from properties (very common in SSC/Banking exams)
2. Finding area from diagonals — especially rhombus and kite
3. Perimeter of regular polygon — given one side or the interior angle
4. Square inscribed in circle (diagonal = diameter) or circle inscribed in square (diameter = side)
5. Angle-based problems — combining polygon angle formulas with parallel line properties

Practice Problems

Q1: The diagonals of a rhombus are 24 cm and 10 cm. Find its side and area.

Q2: Each exterior angle of a regular polygon is 30°. How many sides does it have, and how many diagonals?

Q3: A trapezium has an area of 150 cm². If the parallel sides are in the ratio 3:7 and the height is 12 cm, find the lengths of the parallel sides.

Answers

A1: Half-diagonals are 12 and 5. Side = √(12² + 5²) = √(144 + 25) = √169 = 13 cm (5-12-13 triplet!). Area = ½ × 24 × 10 = 120 cm².

A2: Sides = 360°/30° = 12 sides (dodecagon). Diagonals = 12 × 9 / 2 = 54 diagonals.

A3: Let parallel sides = 3x and 7x. Area = ½(3x + 7x) × 12 = 150. So 60x = 150, x = 2.5. Parallel sides = 7.5 cm and 17.5 cm.