Polygons

A polygon is a closed two-dimensional shape made entirely from straight sides. The name comes from the number of sides:

Sides	Polygon name
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon
9	Nonagon
10	Decagon

Two further classifications:

• Regular: every side and every angle is the same size. A regular triangle is an equilateral triangle; a regular quadrilateral is a square.
• Irregular: not regular — the sides or angles vary.

Quadrilaterals also come with special names:

• Square: all sides equal, four right angles.
• Rectangle: opposite sides equal, four right angles.
• Parallelogram: opposite sides equal and parallel; opposite angles equal.
• Rhombus: all four sides equal; opposite angles equal.
• Trapezium: exactly one pair of parallel sides.
• Kite: two pairs of equal adjacent sides; one pair of equal opposite angles.

Pick a corner of a polygon and draw diagonals to every other corner. A polygon with $n$ sides splits into $(n - 2)$ triangles. Since each triangle's angles sum to $180\degree$, the interior angles of the polygon sum to: $S = (n - 2) \times 180\degree.$

Some useful totals to recognise:

• Triangle ($n = 3$): $180\degree$.
• Quadrilateral ($n = 4$): $360\degree$.
• Pentagon ($n = 5$): $540\degree$.
• Hexagon ($n = 6$): $720\degree$.
• Octagon ($n = 8$): $1080\degree$.

For a regular polygon every interior angle is equal, so divide the total by $n$: $\text{one interior angle} = \frac{(n - 2) \times 180\degree}{n}.$

For an irregular polygon you still know the total — subtract the angles you have been given to find the missing one.

Each exterior angle is the angle you turn through at a corner as you walk around the outside of a polygon. Walk all the way around and you complete one full turn: $\text{sum of exterior angles} = 360\degree.$

This works for every polygon, regular or irregular, no matter how many sides it has.

For a regular polygon every exterior angle is the same: $\text{one exterior angle} = \frac{360\degree}{n}.$

And at every corner the interior and exterior angle sit on a straight line, so: $\text{interior} = 180\degree - \text{exterior}.$

A regular pentagon has $360 \div 5 = 72\degree$ at each corner, so each interior angle is $180 - 72 = 108\degree$.

Working backwards is just as common. Given a regular polygon's interior angle of $144\degree$:

1. Exterior angle $= 180 - 144 = 36\degree$.
2. Number of sides $= 360 \div 36 = 10$.

So the shape is a regular decagon.

Two shapes are congruent if one is an exact copy of the other — same shape, same size. One may be rotated or flipped, but the dimensions match exactly.

For triangles in particular, there are four classic conditions that prove congruence:

• SSS — three sides equal.
• SAS — two sides and the included angle equal.
• ASA — two angles and the side between them equal.
• RHS — Right angle, Hypotenuse and one Side equal (right-angled triangles only).

If you can match a pair of triangles to one of these conditions, they are congruent. Order matters: SSA (or its mirror) is not a reliable test.

Congruent shapes have equal areas and equal perimeters. When you mark them up in a diagram, use identical tick marks on equal sides and identical arcs on equal angles — a clear visual record is half the proof.

Two shapes are similar if one is a scaled copy of the other. Their angles are equal, and their sides are in the same ratio (the scale factor).

The scale factor $k$ tells you how the sizes compare. To find missing lengths in similar shapes:

1. Match up the corresponding sides.
2. Find the scale factor: $k = \dfrac{\text{new length}}{\text{old length}}$.
3. Multiply (or divide) other corresponding lengths by $k$.

So if a small triangle has sides $3, 4, 5$ and a similar triangle has its $3$-side scaled to $6$, then $k = 2$ and the other sides become $8$ and $10$.

All circles are similar. All squares are similar. All equilateral triangles are similar — anything where shape is fixed by the rules.

Congruent shapes are a special case of similar shapes — the scale factor is $1$. Similar shapes that are not congruent must have $k \neq 1$.

Polygon properties and the congruence-vs-similarity distinction matter all through the rest of your maths.

• Pythagoras and trigonometry in IGCSE rely on similar right-angled triangles.
• Circle theorems use isosceles triangles, exterior angles and congruent triangles in their proofs.
• Transformations (reflections, rotations, enlargements) preserve either congruence or similarity.
• Bearings, design and architecture all depend on classifying shapes and using their angle properties.

If a later topic starts to feel slippery, return to these basics: name the polygon, check whether the shapes are congruent or similar, and use the angle rules. They will keep paying you back.