Polygons: Understanding Convex and Concave Shapes

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.

The Definition of 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.

Recognizing Convex Polygons

To help you identify convex polygons, here are a few examples:

These shapes are all considered convex:

• Square
• Rectangle
• Triangle
• Hexagon

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.

The Properties of Convex Polygons

Based on the definition, the properties of convex polygons are:

• All interior angles measure less than 180°
• No inward-pointing vertices
• All diagonals remain inside the polygon
• A line intersecting the polygon will intersect it at 2 distinct points

Convex polygons can also be classified based on the length of their sides and measurement of their angles:

• Equilateral convex polygons have equal side lengths
• Equiangular convex polygons have equal angle measures
• Regular convex polygons have equal side and angle measures

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.

Understanding Concave Polygons

A concave polygon has at least one inward-pointing vertex. Some examples include:

• Star shape
• Crescent shape
• Pentagon with a concave vertex

The properties of concave polygons are:

• At least one interior angle measures more than 180°
• At least one inward-pointing vertex
• At least one diagonal extends outside the polygon
• A line intersecting the polygon may intersect at more than two points

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.

Interior Angles of Concave Polygons

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.

Distinguishing Concave Polygons by Dents

One way to identify concave polygons is by checking for "dents" or inward-pointing vertices. If a polygon has a dent, it is concave.

Diagonals of Concave Polygons

In concave polygons, at least one diagonal may extend outside the figure. In contrast, the diagonals of convex polygons remain entirely within the shape.

Line Intersections in Concave Polygons

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.

How to Differentiate Between Convex and Concave 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 Line Segment Test

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.

The Angle Test

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°.

Key Differences Between Convex and Concave Polygons

To summarize the variations between convex and concave polygons:

• Convex polygons have all interior angles that are less than 180°, while concave polygons have at least one angle that measures more than 180°.
• All vertices in a convex polygon point outward, while concave polygons have at least one vertex pointing inward.
• The diagonals of a convex polygon will always stay within the shape, while concave polygons may have diagonals that intersect the outer boundary.
• A straight line intersecting a convex polygon will do so at only two points, whereas a line intersecting a concave polygon may intersect at more than two points.
• A regular convex polygon has equal sides and angles, while an irregular convex polygon may have varying degrees of sides and angles.

Understanding Convexity in 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 Properties of Convex Polygons

• All interior angles measure less than 180°.
• There are no inward-pointing vertices.
• All diagonals stay within the polygon.
• When intersecting the polygon, a straight line will intersect at two distinct points.

Distinguishing and Differentiating Convex and Concave Polygons

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.