Lesson 63: Polygons

New Concept

Polygons are closed, flat shapes formed by line segments.

What’s next

Next, you’ll learn to identify and name different polygons, from triangles to decagons, based on their number of sides.

Polygons are closed, flat shapes formed by line segments. They are named according to the number of sides they have, and the lengths of the sides may or may not be the same. The key features are being closed, flat, and having straight sides.

A perfect star shape is a polygon because it's flat, closed, and made of only straight lines.
A circle is not a polygon because its edge is a curve, not a collection of straight line segments.
The letter 'C' is not a polygon because it is not a closed shape; there's a noticeable gap.

Think of polygons as the ultimate 2D VIP club for shapes! To get in, a shape must follow three rules: it must be completely flat, all its sides must be straight lines (no curves allowed!), and it must be closed, meaning all lines connect without gaps. It's like a secret fort made entirely of straight sticks!

If a polygon's sides are all the same length and its angles are all the same size, it is called a regular polygon. This means all sides are congruent and all angles are congruent, creating a perfectly symmetrical shape for the number of sides it has.

A square is a regular quadrilateral because all four of its sides are equal and all four of its angles are $90^\circ$.
An equilateral triangle is a regular polygon because it has three equal sides and three equal angles of $60^\circ$.
A standard rectangle is not a regular polygon because even though its angles are equal, its adjacent sides are not.

A regular polygon is the superstar of the shape world, where everything is perfectly balanced. Imagine a shape where every single side is the exact same length and every corner has the exact same angle measurement. No side is longer and no corner is wider—it is pure geometric harmony and fairness. A perfect square is a classic example.

Each corner of a polygon is called a vertex (plural: vertices). A polygon has as many vertices as it has sides. This one-to-one relationship is a fundamental property of all simple polygons, making it easy to determine one count if you know the other.

A pentagon has 5 sides, so it must also have 5 vertices where the sides meet.
An octagon has 8 sides and 8 vertices. A triangle has 3 sides and 3 vertices. So, an octagon has $8 - 3 = 5$ more vertices than a triangle.
A decagon (10 sides) has 10 vertices, while a hexagon (6 sides) has 6 vertices. The decagon has 4 more.

Think of vertices as the 'pointy bits' or corners where the sides of a polygon meet up. They are the joints that hold the entire shape together. The coolest part is that there's a perfect one-to-one match: if you know the number of sides a polygon has, you automatically know it also has that exact same number of vertices.