Area of Polygons — Complete Formula Guide with Practice

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Introduction​

Understanding the area of polygons is one of the most essential skills in geometry. Whether you're a student preparing for exams or someone who wants to refresh their math knowledge, this guide covers everything you need to know — from basic shapes to more complex polygons — with clear formulas, visual explanations, and practice problems.

What Is a Polygon?​

A polygon is a closed 2D shape made up of straight lines. The number of sides determines what type of polygon it is:

Sides	Name
3	Triangle
4	Quadrilateral (Rectangle, Square, etc.)
5	Pentagon
6	Hexagon
n	n-gon

Area Formulas for Each Polygon​

1. Triangle​

Formula: A = ½ × b × h

• b = base
• h = height (perpendicular to base)

Example: A triangle with base = 8 cm and height = 5 cm A = ½ × 8 × 5 = 20 cm²

2. Rectangle​

Formula: A = w × l

• w = width
• l = length

Example: A rectangle with width = 4 cm and length = 9 cm A = 4 × 9 = 36 cm²

3. Square​

Formula: A = a²

• a = side length

Example: A square with side = 6 cm A = 6² = 36 cm²

4. Parallelogram​

Formula: A = b × h

• b = base
• h = perpendicular height

Example: A parallelogram with base = 10 cm and height = 4 cm A = 10 × 4 = 40 cm²

5. Rhombus​

Formula: A = (d₁ × d₂) ÷ 2

• d₁ = first diagonal
• d₂ = second diagonal

Example: A rhombus with diagonals 6 cm and 8 cm A = (6 × 8) ÷ 2 = 24 cm²

6. Trapezoid​

Formula: A = ½ × (a + b) × h

• a = long base
• b = short base
• h = height

Example: Trapezoid with bases 10 cm and 6 cm, height = 4 cm A = ½ × (10 + 6) × 4 = 32 cm²

7. Kite​

Formula: A = (d₁ × d₂) ÷ 2

• d₁, d₂ = the two diagonals

Example: A kite with diagonals 5 cm and 12 cm A = (5 × 12) ÷ 2 = 30 cm²

8. Pentagon (Regular)​

Formula: A = ½ × p × a

• p = perimeter
• a = apothem (distance from center to middle of a side)

Example: Pentagon with perimeter = 30 cm and apothem = 4.1 cm A = ½ × 30 × 4.1 = 61.5 cm²

9. Hexagon (Regular)​

Formula: A = (3√3 ÷ 2) × a²

• a = side length

Example: Hexagon with side = 4 cm A = (3√3 ÷ 2) × 16 ≈ 41.57 cm²

Quick Reference Cheat Sheet​

Shape	Formula	Key Measurements
Triangle	½ × b × h	base, height
Rectangle	w × l	width, length
Square	a²	side
Parallelogram	b × h	base, height
Rhombus	(d₁ × d₂) ÷ 2	diagonals
Trapezoid	½ × (a+b) × h	both bases, height
Kite	(d₁ × d₂) ÷ 2	diagonals
Pentagon	½ × p × a	perimeter, apothem
Hexagon	(3√3÷2) × a²	side

Practice Problems​

Try solving these on your own before checking the answers!

• Problem 1: A triangle has a base of 12 cm and a height of 7 cm. What is its area?
• Problem 2: A rhombus has diagonals of 10 cm and 14 cm. Find the area.
• Problem 3: A trapezoid has parallel sides of 8 cm and 5 cm, with a height of 6 cm. Calculate the area.
• Problem 4: A regular hexagon has a side length of 3 cm. What is its area?
• Problem 5: A kite has diagonals measuring 9 cm and 11 cm. Find the area.

Answers​

Problem 1: A = ½ × 12 × 7 = 42 cm²
Problem 2: A = (10 × 14) ÷ 2 = 70 cm²
Problem 3: A = ½ × (8 + 5) × 6 = 39 cm²
Problem 4: A = (3√3 ÷ 2) × 9 ≈ 23.38 cm²
Problem 5: A = (9 × 11) ÷ 2 = 49.5 cm²

Common Mistakes to Avoid​

• Confusing perimeter with area — perimeter is the total length around a shape, area is the space inside
• Using the slant height instead of perpendicular height — especially common with triangles and parallelograms
• Forgetting to square the units — area is always in cm², m², etc.
• Mixing up diagonals and sides — for rhombus and kite formulas

Summary​

Mastering polygon areas comes down to memorizing the right formula for each shape and identifying the correct measurements. Use the cheat sheet above as a quick reference, and practice regularly with the problems provided.