Counting Sides and Vertices (Corners) - First Grade Mathematics

When mathematicians want to identify and classify any flat shape, they do not guess; they count its parts! Two of the most important parts of any polygon are its straight sides and its vertices. A side is a straight line segment that forms the outer edge of a shape. A vertex (the fancy mathematical word for a corner) is the sharp point where two straight sides meet. In this chapter, you will master the art of counting sides and vertices, discover the magical matching rule that connects them, and avoid common counting mistakes.

What Is a Side?

A side is one of the straight line segments that outline a polygon.

The Sides of a Rectangle:
             Side 1 (Top)
        +-------------------+
Side 4  |                   |  Side 2
(Left)  |                   |  (Right)
        +-------------------+
            Side 3 (Bottom)

Total: 4 Straight Sides!

How to Count Sides Without Recounting:

Place your finger on one side and make a small pencil tick mark across it as you say "one." Trace along the perimeter clockwise, ticking each straight edge: "two, three, four." When you return to your first mark, stop!

What Is a Vertex? (Plural: Vertices)

A vertex is the point or sharp corner where two straight sides intersect. When you have more than one vertex, we say the word VERTICES (say: VER-tih-seez).

The Vertices of a Triangle:
               Vertex 1
                 (o)
                 / \
                /   \
               /     \
             (o)-----(o)
          Vertex 2  Vertex 3

Total: 3 Vertices (Corners)!

Feeling the Vertices

If you cut a triangle or square out of stiff cardboard, run your finger along the edge. The sides feel flat and straight. But when you hit a corner, your finger feels a sharp point! That sharp point is the vertex.

The Magical Polygon Rule: Sides ALWAYS Equal Vertices!

Here is one of the most remarkable discoveries in all of geometry: in every single polygon, the number of sides ALWAYS matches the number of vertices!

The Matching Rule of Polygons:
Polygon         Sides     Vertices     Do They Match?
Triangle          3          3              YES!
Quadrilateral     4          4              YES!
Pentagon          5          5              YES!
Hexagon           6          6              YES!
Octagon           8          8              YES!

The Double-Check Secret

If you count 5 sides on a shape, you know with 100% certainty that it must also have 5 vertices! If you counted 5 sides and only 4 vertices, you know you made a counting mistake and need to check your work.

What About Curved Shapes?

Shapes with curves do not follow polygon rules!

Curved Shapes:
Circle:    0 straight sides,  0 vertices.
Oval:      0 straight sides,  0 vertices.
Crescent:  0 straight sides (curved edges), 2 sharp points (not straight vertices).

Practice Exercises

Exercise 1: Define what a side is, and define what a vertex is. What is the plural word for vertex?

Exercise 2: Count the sides and vertices for each of these shapes: triangle, square, rectangle, and hexagon.

Exercise 3: If a polygon has 5 straight sides, how many vertices must it have? What is the name of a five-sided polygon?

Exercise 4: Explain why a circle has zero vertices.

Solutions and Explanations

Solution to Exercise 1: A side is a straight line segment that forms an outer edge of a 2D shape. A vertex is the corner point where two straight sides meet. The plural word for vertex is vertices.

Solution to Exercise 2: Triangle: 3 sides and 3 vertices. Square: 4 sides and 4 vertices. Rectangle: 4 sides and 4 vertices. Hexagon: 6 sides and 6 vertices.

Solution to Exercise 3: A polygon with 5 sides must have exactly 5 vertices (sides always equal vertices in polygons). A five-sided polygon is called a pentagon.

Solution to Exercise 4: A circle has zero vertices because it is a continuous curved line that never has two straight sides meeting at a sharp corner.