A polygon is a closed shape with three or more straight edges, such as a triangle, square, or rectangle. In this article, we will delve into the concept of convexity in polygons and explore the differences between convex and concave shapes, as well as regular and irregular convex polygons.
A convex polygon is one in which all vertices point outwards. Vertices are points where two sides of a polygon meet. If you need a refresher on polygons, check out our previous article on the topic.
To help you identify convex polygons, here are a few examples:
These shapes are all considered convex:
Convex polygons are prevalent in our daily lives, such as paper and road signs. Even natural formations, like honeycombs, exhibit convexity with their hexagonal shape.
Based on the definition, the properties of convex polygons are:
Convex polygons can also be classified based on the length of their sides and measurement of their angles:
Regular convex polygons with five or more sides are labeled as "regular" before their name. The center of a regular polygon is equidistant from all vertices, making them lie on a circle known as the circumcircle.
In contrast, irregular convex polygons have varying side and angle measurements. An example is a parallelogram.
A concave polygon has at least one inward-pointing vertex. Some examples include:
The properties of concave polygons are:
Now that you understand the difference between convex and concave polygons, you can try to classify shapes based on their convexity. Developing this skill will make geometry and other mathematical concepts easier to understand.
A concave polygon has three or more sides and at least one inward-pointing vertex. In comparison, a convex polygon has all outward-pointing vertices.
One way to identify concave polygons is by checking for "dents" or inward-pointing vertices. If a polygon has a dent, it is concave.
In concave polygons, at least one diagonal may extend outside the figure. In contrast, the diagonals of convex polygons remain entirely within the shape.
Lastly, concave polygons can have intersecting lines that cross at more than two points, while convex polygons will only have two points of intersection. This difference is another way to distinguish between the two types of polygons.
Distinguishing between convex and concave polygons can be determined through a variety of tests that rely on the properties unique to each type of polygon.
The most basic test is the line segment test, where a straight line is drawn between two points within the polygon. If the entire line remains inside the shape without intersecting the outer boundary, the polygon is convex. However, if the line touches the outer boundary at any point, the shape is classified as concave.
In a convex polygon, all interior angles measure less than 180°, while a concave polygon will have at least one angle that measures more than 180°.
To summarize the variations between convex and concave polygons:
What Defines a Convex Polygon?
A convex polygon is a shape where all vertices point outward. It must have a minimum of three sides, but it can have an unlimited number of sides.
Regular Convex Polygons
A regular convex polygon features equal sides and angles, with all vertices lying on a circle. It is both equilateral and equiangular in nature.
The primary contrast between convex and concave polygons is the direction in which their vertices point. In a convex polygon, all vertices point outward, while in a concave polygon, at least one vertex points inward.
