Name: Area Worksheet
Brand: Superprof
SKU: SP-RB-00012841
Rating: 4 (3 reviews)

In geometry, a regular polygon is a two-dimensional closed shape with equal side lengths (equilateral) and equal interior angles (equiangular). Every regular polygon has a distinct geometric centre. The straight-line perpendicular distance from this central point to the midpoint of any outer side is known as the apothem.

Understanding the apothem is a key topic across GCSE and A-Level Mathematics (Edexcel, AQA, OCR). The apothem serves as the height of the internal triangles that construct the polygon, providing the simplest method to calculate the area of complex polygons such as pentagons, hexagons, and octagons.

The best Maths tutors available

What is an Apothem?

Formal Definition:

The apothem (a) of a regular polygon is the line segment drawn from the central point of the polygon perpendicular to the midpoint of one of its sides.

Inradius Connection: The apothem is identical to the radius of the inscribed circle (incircle) that sits perfectly inside the polygon, tangent to all its sides.

Circumradius (R) vs. Apothem (a): The circumradius connects the centre to a vertex, while the apothem connects the centre perpendicularly to the midpoint of a side.

Illustration of what an apothem, circumradius and inradius connection are drawn on a polygon — Image Source: Gianpiero Placidi

Deriving the Apothem Formula for Any n-Sided Polygon

Any regular n-sided polygon with side length s can be partitioned into n identical isosceles triangles radiating from the centre. Bisecting one isosceles triangle forms a right-angled reference triangle with:

Hypotenuse: Circumradius (R)
Adjacent side: Apothem (a)
Opposite side: Half-side length $\left(\dfrac{s}{2}\right)$
Half-central angle: $\alpha = \dfrac{360^\circ}{2n} = \dfrac{180^\circ}{n}$

A. Formula Using Side Length (s)

Using the tangent trigonometric ratio:

\tan\left(\dfrac{180^\circ}{n}\right) = \dfrac{\text{Opposite}}{\text{Adjacent}} = \dfrac{\dfrac{s}{2}}{a}

Rearranging for the apothem a:

a = \dfrac{s}{2\tan\left(\dfrac{180^\circ}{n}\right)}

B. Formula Using Circumradius (R)

Using the cosine trigonometric ratio:

\cos\left(\dfrac{180^\circ}{n}\right) = \dfrac{\text{Adjacent}}{\text{Hypotenuse}} = \dfrac{a}{R} \implies a = R \cos\left(\dfrac{180^\circ}{n}\right)

C. Formula Using Pythagoras' Theorem

When both the circumradius Rand side length s are known:

R^2 = a^2 + \left(\dfrac{s}{2}\right)^2 \implies a = \sqrt{R^2 - \left(\dfrac{s}{2}\right)^2}

Calculating the Area of a Regular Polygon Using the Apothem

Because a regular polygon consists of n congruent triangles, each with base s and height a:

\text{Area of 1 triangle} = \dfrac{1}{2} \times \text{base} \times \text{height} = \dfrac{1}{2} s a

\text{Total Area} = n \times \left(\dfrac{1}{2} s a\right) = \dfrac{1}{2} (n s) a

Since the perimeter is $P = ns$ :

Universal Area Formula for Regular Polygons:

\text{Area} = \dfrac{1}{2} \times \text{Perimeter} \times \text{Apothem} = \dfrac{1}{2} P a

Apothems of Common Regular Polygons

Square (n = 4):The centre splits the square evenly. The apothem is exactly half the side length: $a = \dfrac{s}{2}$
Regular Hexagon (n = 6):A regular hexagon splits into 6 equilateral triangles. Therefore, the circumradius equals the side length (R = s).

a = \sqrt{s^2 - \left(\dfrac{s}{2}\right)^2} = \sqrt{\dfrac{3s^2}{4}} = \dfrac{s\sqrt{3}}{2} = R\cos(30^\circ)

Equilateral Triangle (n = 3):

\alpha = \dfrac{180^\circ}{3} = 60^\circ \implies a = \dfrac{s}{2\tan(60^\circ)} = \dfrac{s}{2\sqrt{3}} = \dfrac{s\sqrt{3}}{6}

Regular Pentagon (n = 5):

\alpha = \dfrac{180^\circ}{5} = 36^\circ \implies a = \dfrac{s}{2\tan(36^\circ)} \approx \dfrac{s}{1.4531} \approx 0.6882 s

Exam Focus & Common Pitfalls

Examiner Tip #1: Hexagon Side Length Equals Circumradius

In exam questions on regular hexagons, remember that each interior central triangle is equilateral. This means the side length always equals the circumradius (s = R).

Examiner Tip #2: Check Calculator Angle Mode

When calculating trigonometric values such as $\tan\left(\dfrac{180^\circ}{n}\right)$ , ensure your calculator is set to Degrees (DEG) mode. If working with radians, use $\tan\left(\dfrac{\pi}{n}\right)$ in Radians (RAD) mode.

Common Pitfall: Confusing the Apothem with the Radius

Apothem (a): Distance to the midpoint of a side (always forms a $90^\circ$ angle with the side).
Radius (R): Distance to a corner / vertex.The apothem is always strictly shorter than the circumradius (a < R).

Practice Questions & Solutions

Score:

Calculate the apothem and area of a square with a side length of 14cm.

Solution

Step 1: Calculate the apothem.

$\text{[math]}$

Step 2: Calculate the perimeter.

$\text{[math]}$

Step 3: Calculate the area using the apothem formula.

$\text{[math]}$

A regular hexagon is inscribed in a circle of radius R = 10cm. Find its exact apothem and calculate its total area.

Solution

Step 1: Identify geometric relationships.

For a regular hexagon, side length s = R = 10cm.

Half-side length is:

$\text{[math]}$

Step 2: Calculate the apothem using Pythagoras' Theorem.

$\text{[math]}$

Step 3: Calculate the total area.

$\text{[math]}$

Find the apothem and area of a regular pentagon with side length s = 8cm, giving your answers to 2 decimal places.

Solution

Step 1: Calculate the half-central angle.

$\text{[math]}$

Step 2: Apply the tangent formula for the apothem.

$\text{[math]}$

Step 3: Calculate the area.

$\text{[math]}$

A regular octagon has an apothem of a = 6cm. Determine its side length s and total area to 2 decimal places.

Solution

Step 1: Determine the half-central angle.

$\text{[math]}$

Step 2: Rearrange for side length s.

$\text{[math]}$

Step 3: Calculate perimeter and area.

$\text{[math]}$

An equilateral triangle has a side length of s = 12cm. Find its circumradius R and apothem a, and verify the ratio $\text{[math]}$

Solution

Step 1: Calculate the apothem a.

$\text{[math]}$

Step 2: Calculate the circumradius R.

$\text{[math]}$

Step 3: Compare ratio.

$\text{[math]}$

(In any equilateral triangle, the apothem is always exactly half the circumradius).

Summarise with AI:

Did you like this article? Rate it!

4.00 (13 rating(s))

Gianpiero Placidi

UK-based Chemistry graduate with a passion for education, providing clear explanations and thoughtful guidance to inspire student success.

Nice emma really precise information.. I appreciate u

Vanessa - Editorial Manager Superprof UK

April 2025

Hey Raptor! Thanks very much for the positive feedback, very much appreciated!

aditi

February 2025

explains everything for a child or a confused student in their ap maths class who is lost in the more detailed mathematics

Vijay Patidar

June 2024

Draw all types of polygon and its area and perimeter class 7th NCERT course

John

September 2023

I’m just curious if the area between the polygon and the circumscribed circle has a name.

7ygugiyv

June 2023

https://www.superprof.co.uk/resources/academic/maths/geometry/plane/orthocenter-centroid-circumcenter-and-incenter-of-a-triangle.html

Sonal patidt

June 2024

Draw all types of polygon and its area and perimeter

Summarise with AI:

Chapters

What is an Apothem?
Deriving the Apothem Formula for Any n-Sided Polygon
Calculating the Area of a Regular Polygon Using the Apothem
Apothems of Common Regular Polygons
Exam Focus & Common Pitfalls
Practice Questions & Solutions

How to Find the Apothem of a Polygon

What is an Apothem?

Deriving the Apothem Formula for Any n-Sided Polygon

A. Formula Using Side Length (s)

B. Formula Using Circumradius (R)

C. Formula Using Pythagoras' Theorem

Calculating the Area of a Regular Polygon Using the Apothem

Apothems of Common Regular Polygons

Exam Focus & Common Pitfalls

Practice Questions & Solutions

Theory

Irregular Polygons

Regular Pentagons

Points, Lines and Planes

Quadrilaterals and Regular Polygons

Rhombus

Diagonals of a Polygon

Concentric Circles

Area and Perimeter of a Triangle

Circular Sectors

Regular Polygons

Similar Triangles

Angles of a Polygon

Parallelograms

Orthocenter, Centroid, Circumcenter and Incenter of a Triangle

Trapezoids

Circumference

Equilateral Triangles

Circles

Quadrilaterals

Circular Segments

Angle Bisectors

Coplanar

Angles of the Triangle

Angles

Intersection of Three Planes

Height of a Polygon

Polygon Rules: The Essential Formulas for Angles & Area

Plane Equation

Line Segments

Points

Medians of a Triangle

Star Polygons

Sheaf of Planes

Lune of Hippocrates

Circumscribed Polygons

Polygons

Rectangle

Right Triangle

Lines

Regular Hexagons

Plane

Triangles

Rhomboid

Perpendicular Bisectors

Intersection of Two Planes

Squares

Area and Perimeter of Polygons

Inscribed Polygons

Sides of A Polygon: Angle Formula Made Simple with Practice Questions

How Do You Find the Apothem? A Clear Step-by-Step Guide

Formulas

Circle Formulas

Plane Formulas

Geometry Formulas

Area and Perimeter Formulas

Perimeter Formulas

Area Formula For All Shapes: Exam Tips and Worked Examples

Triangle Formulas: A Clear Guide to Key Rules

Exercises

Rectangle Problems

Pythagorean Theorem

Plane Problems

Square Problems

Circle Worksheet

Triangle Problems

Area Worksheet

Pythagorean Theorem Word Problems with Answers

Area Word Problems And Worked Solutions

Circle Word Problems And Solutions