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Whether you are studying floor tilings, structural trusses, or geometric proofs, polygons are the fundamental two-dimensional building blocks of geometry. The word "polygon" originates from the Greek words poly (meaning "many") and gonia (meaning "angle").

While basic polygons like equilateral triangles and squares are familiar, higher-tier calculus and geometry problems require a deeper understanding of a polygon's internal anatomy—including its apothems, diagonals, heights, medians, and area mechanics.

In this guide, we will break down the rules of polygons, examine their key geometric components, provide essential formulas, and work through step-by-step practice problems.

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What Defines a Polygon?

A polygon is a flat, two-dimensional (2D) closed plane figure bounded by straight line segments. To qualify as a polygon, a shape must satisfy three strict criteria:

Straight Line Segments: Every side (edge) must be a straight line segment. Curved lines or arcs are not permitted (which is why circles and ellipses are not polygons).
Fully Closed Loop: The sides must connect end-to-end to completely enclose a region of space, leaving no open gaps.
Minimum of Three Sides: A polygon must have at least 3 sides ( $n \ge 3$ ).

names and colour illustrations of the most common polygons — Image Source: Gianpiero Placidi

Classification Rules for Polygons

Polygons are classified into distinct categories based on their internal angles and side lengths:

Convex vs. Concave Polygons

Convex Polygon: Every interior angle is strictly less than $180^\circ$ . If you draw a line segment between any two internal points, the line stays entirely inside the shape. All diagonals lie inside the polygon.
Concave Polygon: At least one interior angle is a reflex angle (greater than $180^\circ$ ). The shape appears to "cave in" on at least one side, causing at least one diagonal to pass outside the boundary of the figure.

Regular vs. Irregular Polygons

Equilateral Polygon: A polygon where all sides are equal in length.
Equiangular Polygon: A polygon where all interior angles are equal in size.
Regular Polygon: A polygon that is both equilateral and equiangular (e.g., a square or an equilateral triangle).
Irregular Polygon: A polygon with sides of differing lengths, angles of differing sizes, or both.

Core Structural Anatomy of a Polygon

To solve advanced coordinate and geometric problems, you must understand the key line segments and geometric features within a polygon.

ANATOMY OF A REGULAR POLYGON

Vertex: Point where two sides meet.
Diagonal: Line connecting two non-adjacent vertices.
Apothem: Perpendicular distance from center to side midpoint.
Height: Perpendicular distance from base to highest apex.
Median: Line from a vertex to the midpoint of the opposite side.

Number of Sides and Vertices (n)

A polygon with n sides always has n vertices (corners) and n interior angles. As the number of sides n approaches infinity, the perimeter and area of a regular polygon approach that of a circle—though a polygon always retains discrete edges.

Diagonals

A diagonal is a straight line segment drawn between any two non-adjacent vertices.

To calculate the total number of diagonals (D) in an n-sided polygon:

From any single vertex, you can draw diagonals to (n - 3) other vertices (you cannot draw a diagonal to the vertex itself or its two immediate neighbors).
Multiplying by n vertices gives n(n - 3), but since every diagonal connects two vertices, this counts each line twice.

D = \dfrac{n(n - 3)}{2}

Height (Altitude)

The height (or altitude) of a polygon is the perpendicular distance measured from a chosen base side to the furthest opposite vertex (or line containing that vertex).

In symmetric polygons with an even number of sides (like a regular hexagon or square), the height is the perpendicular distance between two opposite parallel sides.
In regular polygons with an odd number of sides (like a pentagon), the height is measured from a base side directly to the opposite vertex.

Apothem (a)

The apothem is a line segment drawn from the geometric center of a regular polygon perpendicular to the midpoint of any of its sides.

The apothem is crucial for calculating the area of regular n-gons. For a regular polygon with side length s and n sides, the apothem is given by:

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

Medians

A median is a line segment drawn from a vertex to the midpoint of the opposite side. In triangles, the three medians intersect at a single balance point called the centroid.

Worked Examples

Example 1: Calculating Diagonals

Question: Calculate the total number of diagonals in a regular decagon (10-sided polygon).

Step-by-Step Solution:

Identify the number of sides: n = 10

Apply the diagonal formula: $D = \dfrac{n(n - 3)}{2}$

Substitute n = 10:

D = \dfrac{10(10 - 3)}{2} = \dfrac{10 \times 7}{2} = \dfrac{70}{2} = \mathbf{35 \text{ diagonals}}

Example 2: Area Using Perimeter and Apothem

Question: A regular hexagon has a side length of $6cm$ and an apothem of $3\sqrt{3}\text{ cm}$ (approximately $5.196\text{ cm}$ ). Calculate the exact area of the hexagon.

Step-by-Step Solution:

Calculate the perimeter (P): A hexagon has n = 6 sides

P = n \times s = 6 \times 6 = 36\text{ cm}

Identify the apothem (a):

a = 3\sqrt{3}\text{ cm}

Apply the regular polygon area formula:

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

\text{Area} = \dfrac{1}{2} \times 36 \times 3\sqrt{3} = 18 \times 3\sqrt{3} = \mathbf{54\sqrt{3}\text{ cm}^2 \approx 93.53\text{ cm}^2}

Practice Questions & Answers

Score:

Calculate the total number of diagonals in a 12-sided polygon (a dodecagon).

Solution

Identify n: n = 12

Apply formula:

$\text{[math]}$

A regular pentagon has a perimeter of 40cm and an apothem of 5.5cm. Calculate the area of the pentagon.

Solution

Identify given values: P = 40cm, a = 5.5cm

Apply area formula:

$\text{[math]}$

Find the number of sides of a convex polygon if the total number of diagonals is 54.

Solution

Set up diagonal formula equal to 54:

$\text{[math]}$

Factorise quadratic:

$\text{[math]}$

Select positive solution: n = 12. The polygon has 12 sides.

Summarise with AI:

Did you like this article? Rate it!

4.00 (4 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 Defines a Polygon?
Classification Rules for Polygons
Core Structural Anatomy of a Polygon
Worked Examples
Practice Questions & Answers

Rules for Polygons: Definitions, Types, and Key Parts Explained

What Defines a Polygon?

Classification Rules for Polygons

Convex vs. Concave Polygons

Regular vs. Irregular Polygons

Core Structural Anatomy of a Polygon

Number of Sides and Vertices (n)

Diagonals

Height (Altitude)

Apothem (a)

Medians

Worked Examples

Example 1: Calculating Diagonals

Step-by-Step Solution:

Example 2: Area Using Perimeter and Apothem

Step-by-Step Solution:

Practice Questions & Answers

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