Polygons are closed, two-dimensional shapes made only of straight line segments. In this topic, we look carefully at what makes a shape a polygon and what does not. A shape must meet four important conditions to be a polygon: it must be closed, it must have straight edges, the sides must be non-intersecting, and it must be two-dimensional.

Once we know whether a shape is a polygon, we can classify polygons in more specific ways. Some polygons are regular, which means all sides are the same length and all angles are the same measure. Others are irregular, meaning that not all sides or angles are equal. We will also compare convex and concave polygons. In a convex polygon, all corners point outward. In a concave polygon, at least one part caves inward.

As future teachers, it is important to help students focus on the defining attributes of polygons, not just what shapes “look like.” A student may recognize a triangle or square easily, but still have trouble deciding whether an unfamiliar shape is a polygon. By looking carefully at closed shapes, straight sides, intersections, and dimensions, students build stronger geometric reasoning.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Define polygon using its key attributes.

  • Decide whether a shape is or is not a polygon.

  • Explain why a shape does or does not meet the definition of a polygon.

  • Distinguish between regular and irregular polygons.

  • Distinguish between convex and concave polygons.

  • Classify polygons using precise geometric language.

  • Identify common student misconceptions about polygons.

Key Vocabulary

  • Polygon - A closed, two-dimensional shape made of straight line segments that do not cross.

  • Closed Shape - A shape with no gaps or openings.

  • Straight Edge - A side that is a line segment, not curved.

  • Two-Dimensional - Flat; having length and width but not depth.

  • Non-Intersecting - The sides do not cross over each other.

  • Regular Polygon - A polygon with all sides the same length and all angles the same measure.

  • Irregular Polygon - A polygon where not all sides or angles are equal.

  • Convex Polygon - A polygon with no parts caving inward.

  • Concave Polygon - A polygon with at least one part caving inward.

Classification

  • Regular Polygon - All sides and all angles are equal. - Equilateral triangle, square, regular hexagon

  • Irregular Polygon - Not all sides or angles are equal. - Scalene triangle, irregular pentagon

  • Convex Polygon - No corners cave inward. - Square, regular octagon

  • Concave Polygon - At least one corner caves inward. - Dart-shaped polygon, arrow-shaped polygon

Teacher Connection

Classifying polygons helps students practice using definitions carefully. Rather than asking only, “What is this shape called?” teachers can ask, “How do you know this is a polygon?” or “Which part of the definition does this shape fail?”

This encourages students to justify their thinking with attributes instead of relying on appearance or memory. It also helps prepare students for later geometry, where definitions, examples, and non-examples become very important.

Helpful teacher questions include:

  • Is the shape closed?

  • Are all the sides straight?

  • Do any sides cross?

  • Is this shape flat or three-dimensional?

  • Is this polygon regular or irregular?

  • Is it convex or concave?

  • What part of the definition helped you decide?

Quick Reflection Question

Why is it useful to show students non-examples of polygons, such as open shapes, curved shapes, or shapes with crossing sides? How can non-examples help students understand the definition more clearly?