Learning Materials
Features
Discover
Recalling the concept of polygons, we can say that they are closed shapes with at least three sides, and straight edges. This includes several shapes that we are already familiar with, like triangles, squares, rectangles, and so on. Polygon shapes can be classified based on different aspects, particularly, we will focus on the classification of polygons in terms of their convexity.
Millions of flashcards designed to help you ace your studies
Sign up for free
Achieve better grades quicker with Premium
Geld-zurück-Garantie, wenn du durch die Prüfung fällst
Did you know that StudySmarter supports you beyond learning?
Review generated flashcards
Start learning or create your own AI flashcards
Team Convexity in Polygons Teachers
Convexity in polygons refers to the direction in which the vertices of a polygon are pointing, which can be outwards or inwards.
In this article, we will define what a convex polygon is, and its properties, and we will show you some examples of convex polygons that you can find in the real world. We will also explain the differences between convex and concave polygons, and the concepts of regular and irregular convex polygons.
Depending on their convexity, polygons can be classified as convex or concave. First, let's define what we mean by a convex polygon.
A convex polygon can be defined as a polygon that has all its vertices pointing outwards.
Remember that the vertices of a polygon are the endpoints where two sides of the polygon intersect.
Read more about Polygons if you need to refresh the basics.
Let's see some examples to help you recognize convex polygons more easily.
All the polygons below are convex:
Convex polygons examples - Vaia Originals
We are surrounded by convex polygons in our daily life. For example, a piece of paper (square or rectangle), road signs (triangles, rhombuses or hexagons), and in nature as honeycombs (hexagon), etc.
Examples of convex polygons in our daily life - pixabay.com
Based on their definition, we can define the properties of convex polygons as follows:
All its interior angles measure less than 180°.
Interior angles property of convex polygons - Vaia Originals
There are no dents (vertices pointing inwards).
Dents property of convex polygons - Vaia Originals
All the diagonals of a convex polygon will remain completely inside the polygon, without touching the outside area.
Diagonals property of convex polygons - Vaia Originals
A line intersecting a convex polygon will intersect it at 2 distinct points only. One at the point of entry and the other at the point of exit.
Line intersecting at two points property of convex polygons - Vaia Originals
Based on the length of their sides and the measurement of their angles, convex polygons can be classified as follows:
Equilateral convex polygons are polygons with sides of equal length.
An example of an equilateral convex polygon is a rhombus, as all its sides have the same length.
Equilateral convex polygons example (rhombus) - Vaia Originals
Equiangular convex polygons are polygons with angles of equal measure.
An example of an equiangular convex polygon is a rectangle.
Equiangular convex polygons example (rectangle) - Vaia Originals
Regular convex polygons have sides of equal length and angles of equal measure. This type of convex polygons are both equilateral and equiangular.
Regular polygons with five sides or more are denoted with the word 'regular' preceding the name of the polygon.
Some examples of regular convex polygons are shown below.
Regular convex polygons examples - Vaia Originals
Regular convex polygons also have diagonals of the same length. The centre of a regular polygon is equidistant from all its vertices. This means that all the vertices of a regular polygon will lie on a circle. This circle is known as the circumcircle of that polygon.
Please read Regular Polygons to learn more about this topic.
Irregular convex polygons have sides of different length and angles of different measure.
An example of an irregular convex polygon is a parallelogram.
Irregular convex polygons example (parallelogram) - Vaia Originals
If a polygon is not convex, then it is considered to be concave, but what exactly does that mean?
A concave polygon is a polygon that has at least one of its vertices pointing inwards.
Let's see some examples of concave polygons.
All the polygons shown below are concave.
Concave polygons examples - Vaia Originals
Based of their definition, the properties of concave polygons are as follows:
At least 1 interior angle measures more than 180°.
Interior angles property of concave polygons - Vaia Originals
One or more dents (at least 1 vertex points inwards).
Dents property of concave polygons - Vaia Originals
At least 1 diagonal between two vertices of a concave polygon may touch the outside area.
Diagonals property of concave polygons - Vaia Originals
A line intersecting a concave polygon may intersect it at more than 2 points.
Line intersection property of concave polygons - Vaia Originals
There are several tests that can be used to determine if a polygon is convex or concave. These are based on the properties of convex and concave polygons, and are described below.
There are two types of line test that you can do to check if a polygon is convex or concave.
If you draw a line segment between any two points of the interior of a convex polygon, the whole line segment will remain completely inside the figure without touching the outside area. Otherwise, it is concave.
Identify if the polygons below are convex or concave using the line segment test.
Line segment test example - Vaia Originals
If you extend the sides of a convex polygon, the extended side lines will not cross the interior of the polygon. Otherwise, it is concave.
Identify if the polygons below are convex or concave by extending the sides of the polygons.
Extending the sides test example - Vaia Originals
If you measure the interior angles of a convex polygon, all of them must measure less than 180°. If at least 1 of the interior angles measures more than 180°, then it is a concave polygon.
Identify if the polygons below are convex or concave using the angle test.
Angle test example - Vaia Originals
To help you remember the differences between convex and concave polygons, let's summarize their properties in the table below.
Convex polygons | Concave polygons |
|
|
|
|
|
|
|
|
Sign up for free to gain access to all our flashcards.
What is a convex polygon?
A convex polygon can be defined as a polygon that has all its vertices pointing outwards.
How many sides does a convex polygon have?
A convex polygon needs to have 3 sides or more, and it has no upper limit.
What is the difference between convex and concave polygons?
Concave polygons have inward-pointing vertices whereas a convex polygon will have all outward-pointing vertices.
What is a regular convex polygon?
Regular convex polygons have sides of equal length and angles of equal measure. This type of convex polygons are both equilateral and equiangular. All the vertices will also lie on a circle.
What are the properties of convex polygons?
The properties of convex polygons as follows:
At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified.
Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. She gained her PhD in English Literature from Durham University in 2022, taught in Durham University’s English Studies Department, and has contributed to a number of publications. Lily specialises in English Literature, English Language, History, and Philosophy.
Get to know Lily
Gabriel Freitas is an AI Engineer with a solid experience in software development, machine learning algorithms, and generative AI, including large language models’ (LLMs) applications. Graduated in Electrical Engineering at the University of São Paulo, he is currently pursuing an MSc in Computer Engineering at the University of Campinas, specializing in machine learning topics. Gabriel has a strong background in software engineering and has worked on projects involving computer vision, embedded AI, and LLM applications.
Get to know Gabriel
Vaia 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 moreSave explanations to your personalised space and access them anytime, anywhere!
Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of Vaia.
Already have an account? Log in
Sign up to highlight and take notes. It’s 100% free.
Explanations 
Textbooks 
Exams 
GCSE 
IGCSE 
AS 
A Level 
International A Level 
University Admissions Tests 
Flashcards 
Vaia AI 
Notes 
Study Plans 
Study Sets 
Find a job 
Find a degree 
Student Deals 
Magazine 
Mobile App 
