Polygons

This article will give you an introduction to the different types of polygon and their properties. We will also be looking at how to find the area of polygons and how to find internal and external angles.

Polygons Polygons

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Contents
Table of contents

    Definition of a polygon

    We are all familiar with basic shapes such as triangles and squares. Now, let us look at some more general concept for a certain group of shapes: polygons.

    A polygon is a 2-dimensional shape given that:

    • all sides are straight;
    • there are a minimum of 3 sides;
    • the shape is contained (i.e. the starting point of the first side touches the ending point of the last side);
    • none of the sides cross each other.

    Below are examples of polygons. Observe that these shapes all respect the four conditions for a polygon in its definition.

    Examples of polygons - thinglink.com

    Components of polygons

    It is important to recognise these components of polygons:

    • The sides, sometimes called edges, meet at vertices;

    • The angles within a polygon are called the interior angles;

    • The angles on the outside of the polygon are the exterior angles;

    • All polygons, except triangles, have multiple diagonals (that is, lines between two vertices).

    In the figure below, we can see all these components identified in a polygon.

    Components of a polygon – StudySmarter Original

    What isn’t a polygon?

    A curved shape, or a shape that contains a curve such as a semi-circle which is constructed of one straight line and one curved line, is not a polygon.

    The following are all non-polygons.

    1. This is not a polygon since one side is curved.

    A non-polygon - StudySmarter Originals

    2. This is not a polygon since the sides cross each other.

    A non-polygon - StudySmarter Originals

    3. This is not a polygon since it is not a closed shape.

    A non-polygon - StudySmarter Originals

    In this section of the article, we will be looking at types of polygons. These types are sorted by relationships of the sides of polygons or by the shape of the polygons themselves.

    Regular polygons

    A polygon is regular when all sides and angles within the polygon are equal.

    For example, a square is a regular quadrilateral shape.

    Irregular polygons

    A polygon is irregular when the sides and angles are not equal.

    For example, a rectangle is an irregular quadrilateral shape.

    Convex versus concave polygons

    • A convex polygon is one where all of the vertices point outwards.

    • A concave polygon is one in which at least one vertex points inwards. You are less likely to come across convex polygons at this stage, although they are polygons themselves

    More details on irregular and convex or concave polygons can be found in the article Convexity in Polygons.

    The names of different polygons

    You need to know the following:

    Number of sidesPolygon name
    3Triangles
    4Quadrilateral
    5Pentagon
    6Hexagon
    7Heptagon
    8Octagon

    If a shape has n sides, it will also have n internal angles, and it'll be called n-gon!

    Interior angles

    We all know that a triangle contains180°and therefore the interior angles inside a triangle add up to 180°. But how do we work out how many degrees in all polygons?

    Take a quadrilateral, for example. If you divide a quadrilateral into two shapes along the diagonal, you get two triangles. Since each of those triangles have interior angles adding to 180°, we now know that a quadrilateral has interior angles of 360°.

    We can extend this logic to polygons with even more sides. For example, within a pentagon, you can create 3 triangles using diagonal lines. Within a hexagon, you can create 4 triangles, as demonstrated:

    Decomposition of a hexagon in triangles - StudySmarter Original

    Notice a pattern? For a polygon of n sides, we can create n-2 triangles. Therefore, we have a simple formula for working out the number of interior angles in a polygon:

    Formula for interior angles:(n-2)×180

    What is the sum of the interior angles inside a pentagon?

    (n-2)×180=(5-2)×180=540°

    Taking the definition of a regular polygon, we can now work out the interior angles within any regular polygon. Since all the angles need to be equal, we simply divide the number of interior angles by the number of vertices. For example, a square has interior angles equal to 360/4=90 degrees.

    Exterior angles

    The exterior angles are more straightforward than the interior angles. In all cases, the exterior angles sum to 360°. To calculate the exterior angle of a regular polygon, simply divide 360 by the number of sides, n.

    Formula for exterior angles: 360n

    The following is a regular pentagon. Find x and y.

    Exterior angles of a pentagon - StudySmarter Original

    There are two ways we could go about finding these angles: using either the external or the internal angle formula.

    Internal angle method

    We know from the previous example that there are 540° inside a pentagon since, from the internal angle formula, the sum of internal angles is 540°:

    (n-2)×180=(5-2)×180=540°

    We also know that the pentagon is a regular shape, so each interior angle must be equal:

    5405=108°

    Since there are 180°along a straight line, this means that x and y are as follows:

    x=y=180-108=72°.

    External angle method

    Since there are 5 vertices, there will be 5 equal external angles (including both x and y). Therefore, since the external angles sum to 360°, we know that each angle must be equal to 3605=72°. And so we achieve the same answer as previous: x=y=72°.

    Areas of polygons

    It is helpful to be familiar with the formulae for the areas of common polygons.

    PolygonArea formula
    Triangle12×base×height
    SquareLength2
    RectangleLength×width
    ParallelogramBase×height
    Trapezium12×(sum of lengths of parallel sides/bases)×height
    Rhombus12×(product of diagonals)

    Find the area of the following shape. The lengths are given in centimeters.

    Trapezoid - StudySmarter Original

    The formula is 12×(sum of lengths of parallel sides/bases)×height. We are given the height, 4 cm, and the lengths of the parallel sides, 3 cm and 5 cm. Plugging these into the formula we get:

    Area=12×(3+5)×4=16 cm2

    Polygons - Key takeaways

    • A polygon is a two-dimensional, contained shape with straight sides that meet at vertices
    • Diagonals are straight lines that can be drawn between two vertices
    • Angles in a polygon:
      • The interior angles in a polygon sum to 180×(n-2) where n is the number of sides or vertices
      • The exterior angles of a polygon sum to 360°
    • Regularity of polygons:
      • A polygon is regular when all sides and angles within the polygon are equal
      • A polygon is irregular when the sides and angles are not equal
    • Convexity:
      • A convex polygon is one where all of the vertices point outwards
      • A concave polygon is one in which at least one vertex points inwards
    Frequently Asked Questions about Polygons

    What is a polygon?

    A polygon is a two-dimensional, contained shape with straight sides that meet at vertices.

    What are examples of polygons?

    Some examples of polygons are triangle, square, pentagon and hexagon.

    How to find the area of a polygon?

    Each individual polygon has a formula for its area. It is helpful to be familiar with the formulae for the areas of common polygons.

    What are the types of polygons?

    The different types of polygon include regular or irregular and convex or concave polygons.

    How many sides does a polygon have?

    A polygon must have a minimum of 3 sides. There is no maximum number of sides a polygon can have.

    How to find the exterior angle of a polygon?

    The exterior angles of a polygon sum to 360 degrees, so to find each individual external angle, if the polygon is regular, you need to divide 360 by the number of vertices or sides.

    Test your knowledge with multiple choice flashcards

    All the diagonals of a convex polygon lie inside the polygon

    Polygons are closed shapes with at least three sides, and straight edges.

    A regular polygon has all equal 

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