Polygons: Regular vs. Irregular, Convex vs. Concave

By: Mitch Ryan  | 
Polygons come in all shapes and sizes. Literally. khalus / Getty Images

Whether you're studying up for a math test, helping your child with homework or just trying to brush up before trivia night, learning the basic ins and outs of polygons will serve you well.


What Are Polygons?

Polygons are figures two-dimensional plane figures with straight lines, connecting to create a closed shape with only one boundary. The word "polygon" stems from two words from ancient Greek mathematicians: "poly," meaning many, and "gon," which translates to angles.

Real-life examples of a polygon shape (closed figure with straight sides) include a regular pentagon baseball plate or a regular hexagon stop sign. However, both regular and irregular polygons are found in mathematics and art.


4 Types of Polygons

The number of angles and sides of a polygon gives it its individual name. For example, the prefix for five is "penta," so a five-sided polygon is known as a pentagon. The same goes for hexagons, octagons and so on.

Every polygon falls into four semi-interchangeable categories. Closed shapes with more than three sides and angles can either be convex, concave, regular or irregular polygons.


1. Regular Polygon

A regular (or simple) polygon is a shape with equal angles and multiple sides with equal lengths. Regular polygons can also be categorized as convex polygons because the equal sides canter inward where they connect, creating several v-shaped vertices.

2. Irregular Polygon

An irregular (or complex) polygon has at least one interior angle that does not match the rest. Line segments with different lengths cause this outlier point to stretch the shape, causing non-consecutive vertices where the diagonals meet.

3. Convex Polygon

All convex polygons fall into the regular polygon category. However, the convex label mainly focuses on the interior angles of a polygon. For an n-sided polygon to be considered convex, all the angles equal 180 degrees or less.

4. Concave Polygon

Concave polygons have at least one corner, or exterior angle, that connects within the boundary of the closed shape. This concave appearance can create acute exterior angles, adjacent sides with different lengths and at least one interior angle that measures greater than 180 degrees.


4 Rules For Regular Polygons

Now that you know how different polygons are categorized, you can dig a bit deeper into the nuances of these shapes. Read on to learn some important criteria that every n-sided regular polygon must meet.

1. A Polygon Has at Least Three Sides

To be considered a polygon, the shape's number of sides must be greater than or equal to three — and the same goes for its number of angles. With these rules in mind, an equilateral triangle has the least amount of sides to fulfill all regular polygon prerequisites.


2. All the Sides Are the Same Length

Regular polygons can be theoretically made from millions of straight line segments, but each line segment must be equal in length. A regular quadrilateral square is the most common example, but an isosceles triangle can also be considered regular if the lengths of the other two sides equal the length of base.

3. All Interior Angles Are Equal

All the interior angles of a polygon must be equal for it to be considered regular. Knowing this, you can divide 360 degrees by the number of sides to solve each interior angle measurement.

4. Every Interior Angle Must Be Convex

All the interior angles of a regular polygon must be convex. Convex polygons cannot have an interior angle greater than 180 degrees. This means that all the sides of a regular polygon must meet in v-shaped corners.