Convexity in Polygons

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.

Convexity in Polygons Convexity in Polygons

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Contents
Table of contents

    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.

    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.

    Examples of convex polygons

    Let's see some examples to help you recognize convex polygons more easily.

    All the polygons below are convex:

    Convexity in Polygons Convex polygons examples StudySmarterConvex polygons examples - StudySmarter 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.

    Convexity in Polygons Convex polygons examples in daily life StudySmarterExamples of convex polygons in our daily life - pixabay.com

    Properties of convex polygons

    Based on their definition, we can define the properties of convex polygons as follows:

    • All its interior angles measure less than 180°.

    Convexity in Polygons Interior angles property of convex polygons StudySmarterInterior angles property of convex polygons - StudySmarter Originals

    • There are no dents (vertices pointing inwards).

    Convexity in Polygons Dents property of convex polygons StudySmarterDents property of convex polygons - StudySmarter Originals

    • All the diagonals of a convex polygon will remain completely inside the polygon, without touching the outside area.

    Convexity in Polygons Diagonals property of a convex polygons StudySmarterDiagonals property of convex polygons - StudySmarter 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.

    Convexity in Polygons Line intersection property of convex polygons StudySmarterLine intersecting at two points property of convex polygons - StudySmarter Originals

    Types of convex polygons

    Based on the length of their sides and the measurement of their angles, convex polygons can be classified as follows:

    Equilateral convex polygons

    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.

    Convexity in Polygons Equilateral convex polygons example StudySmarterEquilateral convex polygons example (rhombus) - StudySmarter Originals

    Equiangular convex polygons

    Equiangular convex polygons are polygons with angles of equal measure.

    An example of an equiangular convex polygon is a rectangle.

    Convexity in Polygons Equiangular convex polygons example StudySmarterEquiangular convex polygons example (rectangle) - StudySmarter Originals

    Regular convex polygons

    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.

    Convexity in Polygons Regular convex polygons examples StudySmarterRegular convex polygons examples - StudySmarter 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

    Irregular convex polygons have sides of different length and angles of different measure.

    An example of an irregular convex polygon is a parallelogram.

    Convexity in Polygons Irregular convex polygons example StudySmarterIrregular convex polygons example (parallelogram) - StudySmarter Originals

    If a polygon is not convex, then it is considered to be concave, but what exactly does that mean?

    Concave polygon

    A concave polygon is a polygon that has at least one of its vertices pointing inwards.

    Examples of concave polygons

    Let's see some examples of concave polygons.

    All the polygons shown below are concave.

    Convexity in Polygons Concave polygons examples StudySmarterConcave polygons examples - StudySmarter Originals

    Properties of concave polygons

    Based of their definition, the properties of concave polygons are as follows:

    • At least 1 interior angle measures more than 180°.

    Convexity in Polygons Interior angles property of concave polygons StudySmarterInterior angles property of concave polygons - StudySmarter Originals

    • One or more dents (at least 1 vertex points inwards).

    Convexity in Polygons Dents property of concave polygons StudySmarterDents property of concave polygons - StudySmarter Originals

    • At least 1 diagonal between two vertices of a concave polygon may touch the outside area.

    Convexity in Polygons Diagonals property of concave polygons StudySmarterDiagonals property of concave polygons - StudySmarter Originals

    • A line intersecting a concave polygon may intersect it at more than 2 points.

    Convexity in Polygons Line intersection property of concave polygons StudySmarter OriginalsLine intersection property of concave polygons - StudySmarter Originals

    Tests to differentiate convex and concave polygons

    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.

    Line test

    There are two types of line test that you can do to check if a polygon is convex or concave.

    Line segment

    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.

    Convexity in Polygons Line segment test StudySmarterLine segment test example - StudySmarter Originals

    Extending the sides of the polygon

    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.

    Convexity in Polygons Extending the sides test StudySmarterExtending the sides test example - StudySmarter Originals

    Angle test

    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.

    Convexity in Polygons Angle test example StudySmarterAngle test example - StudySmarter Originals

    Concave and convex polygons

    To help you remember the differences between convex and concave polygons, let's summarize their properties in the table below.

    Convex polygons

    Concave polygons

    • All interior angles measure less than 180°.
    • At least 1 interior angle measures more than 180°.
    • No dents (vertices pointing inwards).
    • One or more dents (at least 1 vertex points inwards).
    • All diagonals of a convex polygon will remain completely inside the polygon, without touching the outside area.
    • At least 1 diagonal between two vertices of a concave polygon may touch the outside area.
    • A line intersecting a convex polygon will intersect it at 2 distinct points only.
    • A line intersecting a concave polygon may intersect it at more than 2 points.

    Convexity in Polygons - Key takeaways

    • Polygons are closed shapes with at least three sides, and straight edges.
    • A convex polygon has all interior angles measuring < 180°.
    • A polygon is concave if at least one of its interior angles measures > 180°.
    • All vertices in a convex polygon point outwards, whereas a concave polygon will have at least one inward-pointing vertex.
    • All the diagonals of a convex polygon will remain completely inside the polygon.
    • A line intersecting a convex polygon will intersect it at 2 distinct points only.
    • A regular convex polygon is a polygon with equal sides and interior angles.
    • An irregular convex polygon have sides of different length and angles of different measure.
    Frequently Asked Questions about Convexity in Polygons

    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:

    • All its interior angles measure less than 180°.
    • There are no dents (vertices pointing inwards).
    • All the diagonals of a convex polygon will remain completely inside the polygon, without touching the outside area.
    • 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.

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