### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Polygons

A polygon is a closed, two-dimensional shape formed by a finite number of straight-line segments joined end-to-end, with each segment called a side.

The two types of polygons are regular polygons and irregular polygons.

A regular polygon is a polygon in which all its sides and angles are equal. Some examples of regular polygons include:

• Equilateral triangles (3 sides)
• Squares (4 sides)
• Regular hexagons (6 sides)

An irregular polygon is a polygon in which not all sides and angles are equal. Some examples of irregular polygons include:

• Scalene triangles (3 sides with all sides and angles different)
• Rectangles (4 sides with equal opposite sides but not all four sides equal)
• Irregular pentagons (5 sides with varying side lengths and angles).

Regular polygons are symmetrical and have rotational symmetry, whereas irregular polygons do not have the same level of symmetry.

## Angle Properties of Polygons

### Sum of interior angles

The sum of interior angles in a polygon is directly related to the number of sides (n) in the polygon. The formula for the sum of interior angles in a polygon is: °

Therefore:

This formula is derived from dividing the polygon into (n – 2) non-overlapping triangles. In each triangle, the sum of interior angles is equal to 180°. The sum of the angles in all these triangles equals the sum of the interior angles of the polygon.

For example, an irregular pentagon (with five sides) is divided into (n − 2) triangles, so it is divided into three triangles.

#### Each interior angle in a regular polygon

A regular polygon is a polygon with all sides and angles equal. To find the measure of each interior angle in a regular polygon, divide the sum of the interior angles (S) by the number of sides (n):

Each interior angle For example, the measure of each interior angle in a regular hexagon (6-sided polygon) is:   ### Sum of Exterior Angles

The sum of exterior angles of a polygon, one at each vertex, is always equal to 360°. This is true for both regular and irregular polygons.

• Remember that this refers to the sum of the exterior angles when only one exterior angle is considered at each vertex

For example, the diagram below shows a regular hexagon. It has six sides of equal length and six interior angles of equal measure. For each vertex of the hexagon, a line is extended from the vertex, creating six exterior angles.

Since the hexagon is regular, all of its exterior angles will be equal, each measuring 60°.

In the second diagram, the six exterior angles each measuring 60° are joined together to form a circle:

This visually demonstrates that the sum of the exterior angles of the hexagon is equal to 360°.