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
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 containsand therefore the interior angles inside a triangle add up to . 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 , we now know that a quadrilateral has interior angles of .
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 triangles. Therefore, we have a simple formula for working out the number of interior angles in a polygon:
Formula for interior angles:
What is the sum of the interior angles inside a pentagon?
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 . To calculate the exterior angle of a regular polygon, simply divide 360 by the number of sides, n.
Formula for exterior angles:
The following is a regular pentagon. Find x and y.
Exterior angles of a pentagon - StudySmarter OriginalThere 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 inside a pentagon since, from the internal angle formula, the sum of internal angles is :
We also know that the pentagon is a regular shape, so each interior angle must be equal:
Since there are along a straight line, this means that x and y are as follows:
.
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 , we know that each angle must be equal to . And so we achieve the same answer as previous: .
Areas of polygons
It is helpful to be familiar with the formulae for the areas of common polygons.
Polygon | Area formula |
Triangle | |
Square | |
Rectangle | |
Parallelogram | |
Trapezium | |
Rhombus | |
Find the area of the following shape. The lengths are given in centimeters.
Trapezoid - StudySmarter OriginalThe formula is . 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:
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 where n is the number of sides or vertices
- The exterior angles of a polygon sum to
- 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
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