Defining a Polygon
There are shapes in geometry that are two dimensional, flat and are made of straight lines. Such shapes have vertices (singular: vertex) that are the points where the straight lines of the boundary of a polygon meet. The straight lines that form the sides of the polygon are also called edges. The most common types of polygons are triangles, squares, rhombuses and rectangles. Polygons may be regular or irregular. Depending on whether the sides of a polygon are equal or not we classify every polygon as being regular or irregular. Every polygon has an interior angle at a specific vertex, and a corresponding exterior angle, and the sum of the exterior and the interior angles is always 180°. In the next section of this article we will look at the definition of a regular polygon.
Defining a Regular Polygon
Let’s first make a comparison between two shapes. A square and another quadrilateral, a shape with four sides.
In these images we see that there are two polygons each with four straight lines as sides, but, as we already know, the square has equal sides and angles. However, the quadrilateral in the second image has unequal sides and unequal angles. This leads us to the definition of a regular polygon. All regular polygons have straight lines as sides, and all the sides of a regular polygon are equal. All the interior angles of a regular polygon are also equal to each other.
An equilateral triangle is the simplest polygon with three equal sides and angles.
[Note: We can keep increasing the number of sides in a regular polygon, and this results in the increase in the number of interior angles.]
Types of Regular Polygons
In this image we see that there are eight different types of regular polygons starting off with an equilateral triangle.
The list of regular polygons are:
- Equilateral triangle – 3 equal sides – each angle is 60°
- Square – equal sides – 4 each angle is 90°
- Pentagon – equal sides – 5 each angle is 108°
- Hexagon – equal sides – 6 each angle is 120°
- Heptagon – equal sides – 7 each angle is 128.57°
- Octagon – equal sides – 8 each angle is 135°
- Nonagon – equal sides – 9 each angle is 140°
- Decagon – equal sides – 10 each angle is 144°
So, we can keep increasing the number of sides to give us regular polygons with equal sides and equal interior angles.
Rapid Recall
Solved Examples
Example 1:
State whether the following statements are true or false.
- All polygons are regular polygons
- Regular polygons do not have equal interior angles
- Irregular polygons have the same number of sides as regular polygons
- Regular polygons have equal exterior angles
- An octagon is not an example of a regular polygon
Solution:
- All polygons are regular polygons – False – as there are irregular polygons along with regular polygons under the category of polygons in geometry.
- Regular polygons do not have equal interior angles – False – as regular polygons have equal sides as well as equal angles.
- Irregular polygons have the same number of sides as regular polygons – True – as every regular polygon has a corresponding irregular polygon with the same number of sides.
- Regular polygons have equal exterior angles – True – as the interior angles of a regular polygon are equal, on the same lines, the exterior angles are also equal as these angles form a pair of linear angles, which add up to 180°.
- An octagon is not an example of a regular polygon – False – as an octagon is in fact a regular polygon with 8 sides, and each interior angle is 135°.
All polygons are closed shapes with sides that are straight lines.
A quadrilateral is a closed two dimensional shape with four sides and four vertices. Quadrilaterals may be regular polygons such as a square, or irregular as in a rectangle or a kite.
Yes, as the number of sides increases, the polygon gradually is converted into a circle with an infinite number of sides.