Area of Regular Polygon Using Apothem and Perimeter

Area of Regular Polygon is calculated from its number of sides, side length and apothem. The main formula is half the product of perimeter and apothem. This page covers the area of regular polygon formula, alternate methods using the circumradius or side length, solved examples, units and useful shortcuts for aptitude questions.

On this page

What Is the Area of a Regular Polygon?

The area of a regular polygon is the region enclosed by a polygon whose sides and interior angles are equal. It is calculated as A = 1/2 × perimeter × apothem.

If the polygon has n equal sides of length s and apothem a, its perimeter is P = ns. Therefore, A = 1/2 × ns × a. The apothem is the perpendicular distance from the centre of the polygon to any side. For a regular pentagon with perimeter 30 cm and apothem 4 cm, the area is 1/2 × 30 × 4 = 60 cm².

Area of Regular Polygon Video Lessons

Watch short topic-wise lessons for quick revision.

15 Lessons
Lesson 1 of 15 Quick Revision

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.

Continue with more lessons and practice in PrepShots.Watch More in App - Start ₹1 Trial →
More Area of Regular Polygon Lessons Scroll to explore →

Area of Regular Polygon Formula & Tricks

Important Formulas

Perimeter of a regular polygon
P = ns

Here, n is the number of sides and s is the length of each side.

Main area formula
A = 1/2 × P × a = 1/2 × nsa

Here, P is the perimeter and a is the apothem.

Area using side length
A = ns² / [4 tan(π/n)]

This formula applies when the number of sides n and side length s are known. The angle in the tangent function is in radians.

Area using circumradius
A = 1/2 × nR² × sin(360°/n)

Here, R is the distance from the centre to any vertex and the angle is measured in degrees.

Apothem from side length
a = s / [2 tan(180°/n)]

This relation gives the apothem of a regular n-sided polygon from its side length.

Interior angle
Each interior angle = (n − 2) × 180° / n

All interior angles of a regular polygon have the same measure.

Quick Tricks

Use perimeter and apothem first

When the apothem is given, do not calculate angles or diagonals. Find the perimeter and apply A = 1/2 × P × a directly.

Example: For a regular octagon with side 5 cm and apothem 6 cm, A = 1/2 × (8 × 5) × 6 = 120 cm².
Divide the polygon into equal triangles

Joining the centre to every vertex divides a regular polygon into n congruent triangles. The total area is n × 1/2 × s × a, which gives A = 1/2 × nsa.

Example: A regular hexagon is divided into 6 equilateral triangles when its circumradius equals its side length.
Recognise the regular hexagon shortcut

A regular hexagon can be divided into six equilateral triangles, so its area is A = 3√3s²/2.

Example: For side 4 cm, area = 3√3 × 4²/2 = 24√3 cm².
Check the unit before finalising

Area uses square units. If the side and apothem are in metres, the answer is in m²; converting a length unit by a factor k changes the area by k².

Example: An area of 2 m² equals 20,000 cm² because 1 m² = 10,000 cm².

Area of Regular Polygon Concepts

Area from Perimeter and Apothem

For any regular polygon, area equals half the product of its perimeter and apothem: A = 1/2 × P × a.

The centre-to-side perpendiculars form n triangles. Each triangle has base s and height a. Adding their areas gives n × 1/2 × s × a = 1/2 × P × a. The apothem must be perpendicular to the side used as the base.

Example: If P = 42 cm and a = 8 cm, then A = 1/2 × 42 × 8 = 168 cm².

Area from Number of Sides and Side Length

When only n and s are known, use A = ns²/[4 tan(π/n)].

The apothem is a = s/[2 tan(π/n)]. Substituting this value into A = 1/2 × nsa produces the side-length formula. This method is useful when the apothem is not provided.

Example: For a square, n = 4. The formula gives A = 4s²/[4 tan(π/4)] = s², which agrees with the usual square-area formula.

Area from Circumradius

If the distance R from the centre to each vertex is known, use A = 1/2 × nR² × sin(360°/n).

Joining the centre to all vertices forms n isosceles triangles. Each triangle has two sides R and central angle 360°/n. Its area is 1/2 × R² × sin(360°/n), and multiplying by n gives the polygon area.

Example: For a regular hexagon with R = 6 cm, A = 1/2 × 6 × 6² × sin 60° = 54√3 cm².

Regular Polygon Properties and Special Cases

A regular polygon has equal sides, equal interior angles, equal central angles, and equal apothems from the centre to its sides.

Each central angle is 360°/n, while each interior angle is (n − 2) × 180°/n. For a regular hexagon, the area is 3√3s²/2. For a regular triangle, the area is √3s²/4. A regular quadrilateral is a square with area s².

Example: A regular hexagon with side 8 cm has area 3√3 × 8²/2 = 96√3 cm².

Area of Regular Polygon Visual Guide

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

Regular polygon divided into equal triangles

Visual guide: Regular polygon divided into equal triangles
number of sides n: 5 side length s: 6 cm apothem a: 4 cm perimeter P: 30 cm
The area of a regular polygon is the sum of triangles formed by joining its centre to each side.

Learn more concepts on PrepShots

Start ₹1 Trial →

Practice Area of Regular Polygon Questions

Practise published questions related to this topic.

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?→

Want more practice questions?

Practice in App →

Area of Regular Polygon Quick Quiz

Attempt 5 questions and check your score instantly.

Quick Revision Notes

Area of Regular Polygon: Revision Points

Use these formulas and properties for quick revision.

  • Perimeter of a regular polygon: P = ns.
  • Main area formula: A = 1/2 × P × a = 1/2 × nsa.
  • Apothem is the perpendicular distance from the centre to any side.
  • Area from side length: A = ns²/[4 tan(π/n)].
  • Area from circumradius: A = 1/2 × nR² × sin(360°/n).
  • Each central angle is 360°/n.
  • Each interior angle is (n − 2) × 180°/n.
  • Regular hexagon area: 3√3s²/2; regular square area: s²; regular triangle area: √3s²/4.

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 Polygon FAQs

What is the area of a regular polygon formula?

The formula is A = 1/2 × perimeter × apothem. If there are n sides of length s and apothem a, then A = 1/2 × nsa.

How do you find the area of a regular polygon without the apothem?

If n and side length s are known, use A = ns²/[4 tan(π/n)]. Alternatively, calculate the apothem using a = s/[2 tan(180°/n)] and then use A = 1/2 × nsa.

What is the area of a regular hexagon with side s?

The area is 3√3s²/2. For s = 4 cm, the area is 24√3 cm².

What is the difference between apothem and circumradius?

The apothem is the perpendicular distance from the centre to a side. The circumradius is the distance from the centre to a vertex, so it is generally longer than the apothem.

How is a regular polygon divided to calculate its area?

Join the centre to every vertex or side endpoint. This forms n equal triangles, each with base s and height a. Their total area is n × 1/2 × s × a.

What is the area of a regular pentagon with perimeter 40 cm and apothem 5.5 cm?

A = 1/2 × 40 × 5.5 = 110 cm².

Continue learning Area of Regular Polygon on PrepShots

Continue on PrepShots