Polygons (Edexcel GCSE Maths): Revision Notes

Polygons

What is a polygon?

A polygon is a shape with multiple straight sides that are connected to form a closed figure. These shapes can be classified into two main types: regular and irregular polygons.

A regular polygon has all sides of equal length and all angles of equal measure. Think of it as a perfectly symmetrical shape where every side and angle is identical.

An irregular polygon has sides of different lengths and angles of different measures. These shapes don't have the same symmetry as regular polygons.

Common polygon names

You need to be familiar with the names of polygons based on their number of sides, particularly those with 3 to 10 sides. Here's what you need to know:

• Triangle (3 sides) - The simplest polygon
• Square (4 sides) - Also called a regular quadrilateral
• Pentagon (5 sides) - Think of the Pentagon building in America
• Hexagon (6 sides) - Like the shape of a honeycomb cell
• Heptagon (7 sides) - Less common but still important
• Octagon (8 sides) - Like a stop sign
• Nonagon (9 sides) - Sometimes called enneagon
• Decagon (10 sides) - The largest you typically need to know

Remember that in a regular polygon, all sides and angles are equal, whilst in an irregular polygon, the sides and angles can be different.

Interior and exterior angles

Understanding polygon angles is crucial for solving geometry problems. There are important formulas you need to learn that apply to different types of polygons.

For any polygon (regular or irregular)

The sum of exterior angles in any polygon is always the same, regardless of the number of sides:

$\text{Sum of exterior angles} = 360°$

The sum of interior angles depends on the number of sides:

$\text{Sum of interior angles} = (n - 2) \times 180°$

Where $n$ represents the number of sides in the polygon.

For regular polygons only

Since all angles are equal in regular polygons, we can find individual angles:

$\text{Each exterior angle} = \frac{360°}{n}$

$\text{Each interior angle} = 180° - \text{exterior angle}$

These formulas work because the interior and exterior angles at each vertex are supplementary (they add up to 180°).

Working with angle problems

Let's look at how to use these formulas in practice. If you're given the interior angle of a regular polygon and need to find the number of sides, follow these steps:

This method works because exterior angles must divide evenly into 360° for a regular polygon to exist.

Key concepts to remember

The reason these formulas work is based on fundamental geometric principles. Any polygon can be divided into triangles by drawing lines from one vertex to all non-adjacent vertices. Since each triangle has angles that sum to 180°, and you can make $(n-2)$ triangles from an $n$-sided polygon, the total interior angle sum becomes $(n-2) \times 180°$.

For exterior angles, imagine walking around the perimeter of any polygon and measuring how much you turn at each corner. By the time you return to your starting point, you will have turned through a complete circle, which is 360°.