Area of Regular Hexagon with Side Length Examples

Area of a regular hexagon is found from its side, apothem or perimeter. A regular hexagon has six equal sides and can be divided into six equilateral triangles. Its area is (3√3/2)a², where a is the side length. This page explains the formulas, derivation, properties and numerical examples used in mensuration problems.

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What Is the Area of a Regular Hexagon?

The area of a regular hexagon with side length a is (3√3/2)a² square units. This follows because the hexagon can be divided into six equilateral triangles of side a.

For a regular hexagon, each interior angle is 120° and all six sides are equal. The apothem, or perpendicular distance from the centre to a side, is (√3/2)a. Using the formula area = 1/2 × perimeter × apothem gives 1/2 × 6a × (√3/2)a = (3√3/2)a². For a side of 4 cm, the area is (3√3/2) × 16 = 24√3 cm².

Area of Regular Hexagon Video Lessons

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Area of a Regular Hexagon

Learn how to calculate the area of a regular hexagon using its side length and the standard mensuration formula, with a clear understanding of the required measurements.

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Area of Regular Hexagon Formula & Tricks

Important Formulas

Using the side length
A = (3√3/2)a²

Here, A is the area and a is the length of each side.

Using perimeter and apothem
A = 1/2 × P × r

P is the perimeter and r is the apothem. For a regular hexagon, P = 6a and r = (√3/2)a.

Apothem of a regular hexagon
r = (√3/2)a

The apothem is the perpendicular distance from the centre to any side.

Using the perimeter
A = (√3/24)P²

Since a = P/6, substituting it into A = (3√3/2)a² gives this formula.

Quick Tricks

Divide the hexagon into six triangles

Join the centre to all six vertices. This creates six equilateral triangles, each with area (√3/4)a². Multiplying by 6 gives A = (3√3/2)a².

Example: If a = 6 cm, A = 6 × (√3/4) × 6² = 54√3 cm².
Find the side from the perimeter first

For a regular hexagon, side length equals perimeter divided by 6. Then apply the side-length formula.

Example: If P = 24 cm, a = 24/6 = 4 cm, so A = 24√3 cm².
Use the square relation for comparison

For two regular hexagons, their areas are proportional to the squares of their side lengths: A₁/A₂ = a₁²/a₂².

Example: If one side is twice the other, its area is 2² = 4 times the other area.

Area of Regular Hexagon Concepts

Derivation from six equilateral triangles

A regular hexagon is made of six equilateral triangles when its centre is joined to all six vertices.

Each triangle has side a and area (√3/4)a². Therefore, the hexagon's area is 6 × (√3/4)a² = (3√3/2)a². This derivation also shows that the distance from the centre to every vertex equals the side length.

Example: For a = 2 cm, each triangle has area √3 cm², so the hexagon has area 6√3 cm².

Area using the apothem

The area of any regular polygon is half the product of its perimeter and apothem: A = 1/2Pr.

For a regular hexagon, P = 6a and r = (√3/2)a. Hence A = 1/2 × 6a × (√3/2)a = (3√3/2)a². This method is useful when the apothem and perimeter are given directly.

Example: If P = 30 cm and r = 5√3/2 cm, then A = 1/2 × 30 × 5√3/2 = 75√3/2 cm².

Finding the area from the perimeter

If the perimeter is P, the side length is P/6 and the area is (√3/24)P².

Substitute a = P/6 into the side formula: A = (3√3/2)(P/6)² = (√3/24)P². The perimeter must be in the same length unit throughout the calculation.

Example: For P = 12 m, A = (√3/24) × 144 = 6√3 m².

Diagonal and radius relationships

In a regular hexagon with side a, the circumradius is a, the shortest diagonal is √3a and the longest diagonal is 2a.

The longest diagonal joins opposite vertices and passes through the centre. The shortest diagonal joins vertices with one vertex between them. These relationships can be used to find the side before applying the area formula.

Example: If the longest diagonal is 14 cm, then a = 14/2 = 7 cm and the area is (147√3)/2 cm².

Area of Regular Hexagon Visual Guide

Use the diagram or chart below to understand the concept clearly.

Regular hexagon divided into six equal triangles

Visual guide: Regular hexagon divided into six equal triangles
Side length (a): a Apothem (r): (√3/2)a Number of equal triangles: 6
A regular hexagon has six equal sides of length a and an apothem r from its centre to any side.

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Practice Area of Regular Hexagon Questions

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1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Area of Regular Hexagon: Revision Points

Use these formulas and relationships for quick calculation.

  • Area = (3√3/2)a², where a is the side length.
  • Perimeter P = 6a.
  • Apothem r = (√3/2)a.
  • Area = 1/2 × perimeter × apothem.
  • Area in terms of perimeter: A = (√3/24)P².
  • A regular hexagon is divided into six equilateral triangles from its centre.
  • Circumradius equals the side length.
  • Shortest diagonal = √3a and longest diagonal = 2a.

Related Topic Explainers

Area of Square → Perimeter of Square → Area of Rectangle → Perimeter of Rectangle → Area of Triangle → Perimeter of Triangle → Heron's Formula → Area of Triangle Using Two Sides and Included Angle →

Area of Regular Hexagon FAQs

What is the area of a regular hexagon with side 5 cm?

A = (3√3/2) × 5² = 75√3/2 cm², approximately 64.95 cm².

What is the area of a regular hexagon with perimeter 36 cm?

The side is 36/6 = 6 cm. Therefore, area = (3√3/2) × 6² = 54√3 cm².

How is the area of a regular hexagon calculated using its apothem?

Use A = 1/2 × P × r, where P is the perimeter and r is the apothem. For a regular hexagon, P = 6a and r = (√3/2)a.

What is the side of a regular hexagon whose area is 24√3 cm²?

Using (3√3/2)a² = 24√3, divide by √3 to get (3/2)a² = 24. Thus a² = 16 and a = 4 cm.

What is the difference between the area of a regular hexagon and its perimeter?

Area measures the enclosed surface in square units, while perimeter measures the boundary length in linear units. They cannot be directly subtracted or compared as equal quantities.

If the side of a regular hexagon increases by 10%, how does its area change?

Area depends on the square of the side. The new area is (1.1)² = 1.21 times the original, so it increases by 21%.

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