Polygons

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TABLE OF CONTENTS

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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.

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

Types of polygons

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 sides	Polygon name
3	Triangles
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon

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 contains $180 °$ 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: $(n - 2) \times 180$

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

$(n - 2) \times 180 = (5 - 2) \times 180 = 540 °$

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: $\frac{360}{n}$

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) \times 180 = (5 - 2) \times 180 = 540 °

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

$\frac{540}{5} = 108 °$

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 $\frac{360}{5} = 72 °$ . 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.

Polygon	Area formula
Triangle	$\frac{1}{2} \times b a s e \times h e i g h t$
Square	$L e n g t h^{2}$
Rectangle	$L e n g t h \times w i d t h$
Parallelogram	$B a s e \times h e i g h t$
Trapezium	$\frac{1}{2} \times (s u m o f l e n g t h s o f p a r a l l e l s i d e s / b a s e s) \times h e i g h t$
Rhombus	$\frac{1}{2} \times (p r o d u c t o f d i a g o n a l s)$

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

Trapezoid - StudySmarter Original

The formula is $\frac{1}{2} \times (s u m o f l e n g t h s o f p a r a l l e l s i d e s / b a s e s) \times h e i g h t$ . 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:

$A r e a = \frac{1}{2} \times (3 + 5) \times 4 = 16 {cm}^{2}$

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 $180 \times (n - 2)$ 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

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

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

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

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

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

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.

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What two conditions are needed for two polygons to be similar?

They must have the same number of sides and the same angles at corresponding vertices.

Are all equilateral triangles similar? Yes or no.

Yes

Are all isoceles triangles similar?

No.

Are all similar polygons congruent? Yes or no.

No.

Are all congruent polygons similar? Yes or no.

Yes.

What is the type of relationship between corresponding sides on two similar polygons?

Their lengths are directly proportional.

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Polygons