Welcome to the World of Polygons!

In this chapter, we will explore shapes that make up the world around us—from the honeycomb patterns in a beehive to the floor tiles in your classroom. We call these shapes polygons. By the end of these notes, you will know how to calculate the angles inside and outside any polygon and even how to tile them together!

1. What is a Polygon?

A polygon (多邊形) is a flat, closed shape made of three or more straight line segments.

  • Sides: The straight lines that form the shape.
  • Vertices: The "corners" where two sides meet.
  • Interior Angles: The angles inside the polygon.

Regular Polygons (正多邊形)

A polygon is called a regular polygon if:

  1. All its sides are equal in length.
  2. All its interior angles are equal in size.

Example: An equilateral triangle is a regular polygon with 3 sides. A square is a regular polygon with 4 sides.

Quick Tip: If a shape has curved sides (like a circle) or is open, it is not a polygon!

2. Sum of Interior Angles

Every polygon can be divided into triangles by drawing lines from one vertex to the others (called diagonals). Since the sum of angles in one triangle is \(180^\circ\), we can find the total sum for any polygon.

The Formula

For any polygon with \(n\) sides, the sum of interior angles is:

Sum \( = (n - 2) \times 180^\circ\)

Why \(n - 2\)?
Don't worry if this seems tricky! Just remember: if you have a 4-sided shape (quadrilateral), you can split it into \(4 - 2 = 2\) triangles. Since each triangle has \(180^\circ\), the total is \(2 \times 180^\circ = 360^\circ\).

Interior Angle of a Regular Polygon

Because all angles in a regular polygon are equal, you can find the size of one interior angle by dividing the sum by the number of sides \(n\):

Each Interior Angle \( = \frac{(n - 2) \times 180^\circ}{n}\)

Key Takeaway: To find the sum of angles, just count the sides, subtract 2, and multiply by \(180^\circ\).

3. Sum of Exterior Angles

If you "extend" the sides of a polygon, you create exterior angles. An important rule for any convex polygon (a polygon where all "corners" point outwards) is:

The sum of exterior angles is ALWAYS \(360^\circ\).

It doesn't matter if the polygon has 3 sides or 100 sides; if you walk all the way around the outside, you make one full turn, which is \(360^\circ\).

Exterior Angle of a Regular Polygon

For a regular polygon with \(n\) sides:

Each Exterior Angle \( = \frac{360^\circ}{n}\)

Did you know? At any vertex, the Interior Angle + Exterior Angle = \(180^\circ\) because they lie on a straight line. (This relates back to your Angles and Parallel Lines chapter!)

Common Mistake: Students sometimes forget that the exterior angle sum is fixed at \(360^\circ\) and try to use a formula with \(n\). Remember: Exterior Sum is always 360!

4. Tessellation (Non-foundation Topic)

Tessellation (鑲嵌) is the covering of a flat surface using one or more geometric shapes with no overlaps and no gaps.

Which shapes can tessellate?
  • Triangles and Quadrilaterals: Any triangle or any quadrilateral (even irregular ones!) can tessellate the plane.
  • Regular Polygons: Only three regular polygons can tessellate by themselves:
    • Equilateral triangles
    • Squares
    • Regular hexagons (like a honeycomb!)

Why? For shapes to fit together perfectly at a point, the sum of the angles meeting at that vertex must be exactly \(360^\circ\).

5. Geometric Construction (Non-foundation Topic)

Using only a compass and a straightedge (a ruler without marks), we can construct perfect regular shapes.

Constructing an Equilateral Triangle

  1. Draw a line segment \(AB\).
  2. Place the compass point at \(A\) and draw an arc with radius \(AB\).
  3. Place the compass point at \(B\) and draw another arc with the same radius.
  4. Where the arcs cross is point \(C\). Connect \(AC\) and \(BC\).

Constructing a Regular Hexagon

A regular hexagon is special because its side length is equal to the radius of the circle it fits inside! By drawing a circle and "stepping" the compass around the edge using the same radius, you can mark the 6 vertices perfectly.

Summary Checklist

Quick Review:

  • Regular Polygon: All sides and angles equal.
  • Interior Sum: \((n - 2) \times 180^\circ\).
  • Exterior Sum: Always \(360^\circ\).
  • One Exterior Angle (Regular): \(\frac{360^\circ}{n}\).
  • Tessellation: Tiling with no gaps (Triangles, Squares, and Hexagons work!).

Keep practicing! Polygons are the building blocks of geometry. If you find the formulas confusing, try drawing a square (\(n=4\)) and testing the formula—it will always give you \(360^\circ\)!