Regular Polygon

A polygon can be defined as a closed shape with straight edges. Since they need to be closed, they'll always have three or more sides and having straight edges removes circles and ovals from this category. Polygons can be classified on a lot of different criteria, from the number of sides to their interior and exterior angles.  

Regular Polygon Regular Polygon

Create learning materials about Regular Polygon with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Table of contents

    The criteria that determine whether a shape is convex or concave is the magnitude of interior angles.

    If all the interior angles are less than 180° each, then the shape is classified as convex. Whereas if any one of the interior angles is greater than 180°, then the shape is concave. Convex polygons are further classified into regular or irregular polygons depending on the length of the sides and interior angles.

    In this article, we go through what a regular polygon is, its properties, and a few examples.

    What is a Regular polygon?

    A regular polygon has sides of equal length and equal interior angles.

    Examples of regular polygons are equilateral triangles, squares, rhombuses, and so on.

    A polygon will also have diagonals of the same length. Regular polygons are mostly convex by nature. On the other hand, concave regular polygons are sometimes star-shaped. We will be discussing the properties of regular convex polygons in detail.

    Regular polygon properties

    Circumcircle and Incircle

    There are two important circles that can be drawn on a regular polygon.

    • The circumcircle lies outside the convex regular polygon and passes through all its vertices. The radius of the circumcircle is the distance from the center of the polygon to any of its vertex.
    • The incircle passes through the mid-point of all the sides of the polygon and lies inside the regular polygon. The radius of the incircle is the distance between the center and a midpoint of any side. This distance is also called the apothem of the polygon.

    These properties of an incircle, circumcircle, and apothem can only be found in regular polygons.

    Regular Polygon Properties of a regular polygon StudySmarterCircumcircle, incircle and apothem,

    So what can we do using these properties?. One interesting application is being able to calculate the area of a regular polygon using the apothem. Any regular polygon can be broken down into triangles, Combining this with the apothem we can estimate the areas of any regular polygon of side N.

    Calculate the area of a hexagon with side s and apothem I.

    Regular Polygon Area of a regular polygon StudySmarterArea of a regular polygon,


    Divide the hexagon into six triangles as shown in the image below. We observe the following

    • The base of the triangle is equal to the side of the polygon (s).
    • The height of the triangle is nothing but the apothem of the polygon (l).

    To get the area of a polygon all we need is to calculate the area of one triangle and multiply it by the number of sides.

    Therefore, the Area of the hexagon = 6 × 12 X s × l

    Regular polygon examples

    Regular polygons with 3 sides are called equilateral triangles, 4 sides are called squares. Regular polygons with more than four sides are denoted with a 'regular' preceding the name of the polygon. For example, a pentagon with equal sides and angles is called a regular pentagon. Below are a few examples of a regular (equiangular) convex polygon.

    Regular Polygon Regular polygon examples StudySmarterRegular polygon examples,

    Formulas for regular polygons

    Regular polygons have a few interesting properties associated with each of their attributes. We look at them in the following sections.

    Exterior angles

    At any vertex of a polygon, there are 2 angles the interior and exterior. The exterior angle is obtained by the angle between an extended edge and its consecutive edge.

    Regular Polygon Exterior angles of a regular polygon StudySmarterExterior angles of a regular polygon,

    In a regular convex polygon, the sum of all exterior angles is always 360° It can also be written as,

    =360/N, where N is the number of sides and is the exterior angle.

    Interior Angles

    The interior angles are formed between two adjacent sides of a polygon. The sum of interior angles of a polygon will depend on the number of sides that it has. For example, all triangles will have a total sum of 180°, quadrilaterals will have a sum of 360°, and so on. But what about a polygon with hundred sides.

    Regular Polygon Interior angles of a regular polygon StudySmarterInterior angle of a regular polygon,

    The sum of the interior and exterior angles at a vertex is always equal to 180°. Using this relation we can derive a general equation that can be used to find the interior angles of any polygon by having the number of sides.

    Interior Angle = 180° Exterior AngleWe already know the Exterior angle = 360°N, soInterior Angle = 180° 360°NInterior Angle= 180° 360°N = (N × 180°N) (2 × 180°N) = (N2) × 180°NTherefore, Interior angle= (N2) × 180°NAnd Sum of all interior anlgles=(N2) × 180°

    Calculate the exterior and interior angles and the sum of all interior angles for a regular decagon.

    Regular Polygon Angles of regular decagon StudySmarterAngles of a regular decagon,


    We know that, Exterior angle = 360°N=360°10=36°

    Similarly, Interior angle = 180°-exterioir angle=180°-36°=144°

    Therefore sum of all interior angles = N X 144° = 10 X 144° = 1440°

    Diagonals of a Convex polygon

    In a polygon with more than 3 sides, a diagonal is a line segment between any two non-consecutive points. Unlike the concave polygons, the diagonals of a convex polygon will always lie inside the figure. If a polygon has 'N' sides, then the number of diagonals is equal to:

    Number of diagonals=N(N-3)2.

    Calculate the number of diagonals in a heptagon.


    Applying the formula we get

    Number of diagonals=N(N-3)2=7(7-3)2 = 14 diagonals

    Diagonals of a regular pentagon,

    We get a total of 14 diagonals, which are shown in the figure above.

    This brings us to the end of this article. Let us refresh what we've learned so far.

    Regular Polygons - Key takeaways

      • A polygon with equal sides and interior angles is called a regular polygon.
      • All the diagonals of a regular polygon are equal in length.
      • The circumcircle passes through all the vertices of a regular polygon. The radius of the circumcircle is called the circumradius.
      • The Incircle passes through the midpoints of each side. The radius of the incircle is called the apothem of the polygon.
      • Every regular polygon can be broken down into smaller triangles which can be used to calculate their area.
      • All the vertices of a regular polygon are equidistant from its center.
      • The sum of exterior angles in a regular polygon is always equal to 360°.
      • The sum of all interior angles for a regular polygon is given by (N2) × 180°
      • The number of diagonals for a polygon with 'N' sides is given byN(N-3)2
    Frequently Asked Questions about Regular Polygon

    What is a regular polygon?

    A polygon with equal sides and interior angles is called a regular polygon.

    How many sides does a regular polygon have?

    The minimum number of sides for a regular polygon is 3 ( Equilateral triangle), it has no upper limit.

    What is an example of a regular polygon?

    Equilateral triangles, squares, rhombuses are examples of regular polygons.

    How to find the area of a regular polygon?

    The area of a regular polygon can be calculated by dividing it into triangles. To get the area of the whole polygon, just add up the areas of all the little triangles.

    What shape are regular polygons?

    Regular polygons will have different shapes depending on the number of sides.

    Discover learning materials with the free StudySmarter app

    Sign up for free
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Math Teachers

    • 6 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner