Properties of Polygons and Symmetry

What is a Polygon?

You can easily draw a continuous curve to make a circle, but drawing a polygon requires stopping and changing direction at every corner. This is because polygons are strictly built from straight lines, making their properties predictable and precise.

A polygon is a flat, 2D closed shape formed by three or more straight line segments. These line segments are called sides (or edges), and the corners where they meet are called vertices. A circle is strictly not a polygon because it does not have straight sides.

Polygons can be classified by their inside angles. A convex polygon has all interior angles less than $180^\circ$. By contrast, a concave polygon has at least one reflex angle (greater than $180^\circ$), such as a four-sided arrowhead shape.

Perpendicular Lines and Right Angles

When two straight lines meet at exactly $90^\circ$, they are perpendicular and form a right angle.

In geometry diagrams, a right angle is identified by a small square symbol in the corner. It is rarely labelled with the text "$90^\circ$", so you must recognise this square symbol immediately.

Naming and Regularity

Polygons are named based on their number of sides. At Foundation level, you are expected to recognise and name a triangle (3 sides), quadrilateral (4), pentagon (5), hexagon (6), octagon (8), and decagon (10).

A shape's properties determine if it is a regular polygon or an irregular polygon. To be strictly classified as a regular polygon, a shape must have sides of equal length (equilateral) AND interior angles of equal size (equiangular).

Irregular polygons have sides of different lengths and/or angles of different sizes. For example, a rectangle has four $90^\circ$ angles (equiangular) but unequal sides, making it irregular. A rhombus has four equal sides (equilateral) but unequal angles, so it is also irregular.

Symmetry Properties

A line of symmetry (or reflective symmetry) is a straight line that divides a shape into two identical halves that are mirror images of each other. The order of rotational symmetry is the number of times a shape looks identical to its starting position during one full $360^\circ$ rotation about its centre of rotation.

For any regular polygon with $n$ sides, there are exactly $n$ lines of symmetry, and it has rotational symmetry of order $n$. For example, a regular hexagon has 6 lines of symmetry and rotational symmetry of order 6.

Be highly careful with quadrilaterals, as they often trick students. A square has 4 lines of symmetry, but a rectangle and a rhombus only have 2. A parallelogram has 0 lines of symmetry, but it still maintains rotational symmetry of order 2.

Interior and Exterior Angles

An interior angle is the angle inside the polygon at one of its vertices. An exterior angle is the angle between one side of a polygon and the extended straight line of the adjacent side.

Together, an interior and exterior angle sit on a straight line, meaning they always sum to $180^\circ$.

Worked Examples

Example 1: Finding the smallest angle of rotation Calculate the smallest angle a regular hexagon must be rotated to look identical to its starting position.

Step 1: State the formula for the smallest angle of rotation.

$$\text{Smallest Angle} = \frac{360^\circ}{\text{Order of Rotational Symmetry}}$$

Step 2: Identify the order of rotational symmetry for a regular hexagon.

• A regular hexagon ($n=6$) has rotational symmetry of order 6.

Step 3: Substitute and calculate.

• $\text{Angle} = \frac{360^\circ}{6} = 60^\circ$