Classifying 2D Shapes and Polygons - Third Grade Mathematics

Two-dimensional shapes are flat geometric figures that have length and width, but no depth or thickness. From the stop signs on our streets to the tiles on kitchen floors, polygons define the patterns of our built and natural environments. In this chapter, you will discover what qualifies a shape as a true polygon, learn the names of polygons based on their number of sides and angles, and classify triangles and quadrilaterals by their geometric attributes.

What Is a Polygon?

A polygon is a two-dimensional, closed shape made exclusively of straight line segments that connect end-to-end without crossing each other.

To be a Polygon, a shape MUST:
1. Be 2-dimensional (completely flat).
2. Be closed (no openings; starting point connects to ending point).
3. Have straight sides (no curved lines!).
4. Have sides that do not cross or intersect.

Polygon Examples:
Triangle        Rectangle       Hexagon
   /\           +-------+          /\
  /  \          |       |         /  \
 /____\         +-------+        +----+
                                   \  /
                                    \/

NOT Polygons:
Circle (curved side)     Open shape (gap!)      Figure 8 (sides cross!)
     (   )                     |                  \   /
    (     )               +----+                   \ /
     (   )                                          X
                                                   / \

Naming Polygons by Their Sides and Angles

Polygons are classified and named according to how many straight sides (and vertices) they have. In every polygon, the number of sides is always equal to the number of vertices (corners)!

+-----------------------+-----------------+-------------------+
| Polygon Name          | Number of Sides | Number of Angles  |
+-----------------------+-----------------+-------------------+
| Triangle              | 3 sides         | 3 angles          |
| Quadrilateral         | 4 sides         | 4 angles          |
| Pentagon              | 5 sides         | 5 angles          |
| Hexagon               | 6 sides         | 6 angles          |
| Octagon               | 8 sides         | 8 angles          |
| Decagon               | 10 sides        | 10 angles         |
+-----------------------+-----------------+-------------------+

Memory tips:
- Tri- means 3 (tricycle = 3 wheels).

- Quad- means 4 (quadruplets = 4 babies).

- Penta- means 5 (the Pentagon building in Washington D.C. has 5 sides).

- Hex- means 6 (notice the 'x' in hex and six!).

- Oct- means 8 (octopus = 8 tentacles).

- Dec- means 10 (decade = 10 years).


Exploring Triangles (3-Sided Polygons)

Triangles can be classified by their angles:
- Right Triangle: Contains exactly one right angle (square corner).

- Acute Triangle: All three angles are acute (smaller than square corners).

- Obtuse Triangle: Contains one obtuse angle (wider than a square corner).


Right Triangle:           Acute Triangle:         Obtuse Triangle:
    |\                         /\                     \
    | \                       /  \                     \
    |  \                     /    \                     \_____
    +---\                   /______\

Regular vs. Irregular Polygons

  • Regular Polygon: A polygon where ALL sides have the exact same length and ALL angles have the exact same measure (for example, an equilateral triangle or a square).
  • Irregular Polygon: A polygon with sides of different lengths or angles of different sizes.

Even if a hexagon has sides of completely different lengths, as long as it has 6 straight closed sides, it is still a hexagon!

Chapter Practice Exercises

  1. Tell whether each shape is a polygon. If not, explain why: a. A shape with 4 straight sides and 4 square corners b. A circle c. A shape with 3 straight sides that does not close at one corner d. A 5-sided flat figure with straight sides
  2. Name the polygon with: a. 3 sides b. 8 sides c. 6 sides d. 5 sides e. 4 sides
  3. What is the name of a road stop sign shape? How many sides and vertices does it have?
  4. Can a polygon have 4 sides and only 3 angles? Explain why or why not.
  5. Draw or describe a right triangle. What special angle must it contain?

Solutions and Step-by-Step Answers

  1. Polygon tests: a. Yes, it is a polygon (specifically, a rectangle or square). b. No, a circle has curved boundaries, not straight line segments. c. No, a polygon must be a completely closed figure. d. Yes, it is a polygon (specifically, a pentagon).
  2. Polygon names: a. 3 sides -> Triangle. b. 8 sides -> Octagon. c. 6 sides -> Hexagon. d. 5 sides -> Pentagon. e. 4 sides -> Quadrilateral.
  3. A stop sign is an octagon. It has 8 straight sides and 8 vertices (corners).
  4. No. In every polygon, the number of sides is always equal to the number of angles and vertices. A 4-sided polygon always has exactly 4 angles.
  5. A right triangle has 3 straight sides and contains exactly one right angle (a 90-degree square corner).