Properties of Polygons and Symmetry

Recognising a right angle

A right angle is an angle of exactly $90^\circ$. On a diagram it is normally shown by a small square placed inside the angle. This marking supplies mathematical information: even if the drawing is not accurately to scale, the marked angle is still $90^\circ$.

Two lines that meet at a right angle are perpendicular, but the focus here is the angle itself. An angle which merely looks square is not necessarily a right angle; there must be a marking, a measurement or enough stated information to prove it.

What makes a shape a polygon?

A polygon is a closed, flat shape made entirely from straight sides. Each side meets the next at a vertex. A triangle, quadrilateral and pentagon are polygons because they are closed and have three, four and five straight sides respectively.

This definition gives a useful identification test:

1. Is the shape two-dimensional?
2. Is its boundary closed, with no gap?
3. Is every part of the boundary straight?

If any answer is no, the shape is not a polygon. A circle is not a polygon because its boundary is curved, while an open chain of straight segments is not a polygon because it does not enclose a region.

Polygons may be convex, with every interior angle less than $180^\circ$, or concave, with at least one reflex interior angle. A concave polygon is still a polygon: the inward “dent” does not change the fact that its boundary is closed and straight-sided.

Regular and irregular polygons

A polygon is regular only when it satisfies two conditions at the same time:

• all its sides have equal length;
• all its interior angles have equal size.

A square is therefore a regular quadrilateral. By contrast, a non-square rhombus has equal sides but unequal angles, while a non-square rectangle has equal angles but not all sides equal. Each fails one of the two conditions, so neither is regular.

Marks on a diagram help you decide what is known. Matching dashes on sides show equal lengths, matching arcs show equal angles and a small square shows a right angle. Do not assume equality from appearance alone.

Reflection symmetry

A shape has reflection symmetry, also called line symmetry, if reflection in a line maps the whole shape exactly onto itself. The line is a line of symmetry or mirror line. One way to test it is to imagine folding the shape along the proposed line: corresponding points on the two halves must coincide.

The line may be vertical, horizontal or diagonal. For example, a square has four lines of symmetry: two join the midpoints of opposite sides and two run along its diagonals. A non-square rectangle has two, whereas a general parallelogram has none.

Rotational symmetry

A shape has rotational symmetry when it matches its starting appearance after being turned through an angle greater than $0^\circ$ but less than $360^\circ$ about its centre.

The order of rotational symmetry is the number of matching positions during one complete $360^\circ$ turn, including the final return to the starting position. A non-square rectangle matches after $180^\circ$ and again at , so its rotational order is . A general parallelogram also has order , despite having no lines of symmetry.

Every shape has rotational order at least $1$ because it matches itself after $360^\circ$. Order $1$ is normally described as having no rotational symmetry, since there is no match before the full turn.

Symmetry of regular polygons

A regular $n$-sided polygon has:

• $n$ lines of symmetry;
• rotational symmetry of order $n$.

For example, an equilateral triangle has three lines of symmetry and rotational order $3$; a square has four lines and order $4$; and a regular hexagon has six lines and order $6$. Its successive matching turns are separated by $\frac{360^\circ}{n}$.

This rule depends on the polygon being regular. An irregular hexagon does not automatically have six lines of symmetry or rotational order $6$ merely because it has six sides.

Identifying properties systematically

When classifying a diagram, deal with its properties in a fixed order. First check whether it is a polygon. Then use side and angle markings to decide whether it is regular. Finally, test reflection and rotation separately. This avoids assuming that a shape with rotational symmetry must also possess a mirror line, or that equal sides alone make a polygon regular.