Shapes and Polygons

A 2D shape is a flat shape — it has length and width but no thickness. A drawing on paper is 2D.

A polygon is a special kind of 2D shape: it is closed (the sides join all the way round) and made only of straight sides. A circle is not a polygon because it is curved. An open zig-zag is not a polygon because it does not join up.

The name of a polygon comes from its number of sides:

Useful names to remember: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9) and decagon (10).

A polygon is regular when every side is the same length and every angle is equal — a square and a regular hexagon are good examples. If the sides or angles are different, it is irregular.

Every triangle has 3 sides and 3 angles, but they come in different families. You can sort them two ways.

By their sides:

• Equilateral — all 3 sides equal, and all 3 angles equal ($60\degree$ each).
• Isosceles — exactly 2 sides equal, and the 2 angles opposite them are equal too.
• Scalene — all 3 sides different, and all 3 angles different.

By their angles:

• Acute — all 3 angles are less than $90\degree$.
• Right-angled — has exactly one $90\degree$ angle (shown by a small square).
• Obtuse — has one angle bigger than $90\degree$.

A triangle can have a name from each list — for example, a triangle can be both isosceles and right-angled. On a diagram, look for tick marks that show equal sides and a small square that shows a right angle.

A quadrilateral is any shape with 4 straight sides. The family has several members, each with its own special properties.

Here is what makes each one special:

• Square — 4 equal sides, 4 right angles, 2 pairs of parallel sides.
• Rectangle — opposite sides equal, 4 right angles, 2 pairs of parallel sides.
• Parallelogram — opposite sides equal and parallel, opposite angles equal.
• Rhombus — 4 equal sides, opposite sides parallel, like a "pushed-over" square.
• Trapezium — exactly one pair of parallel sides.
• Kite — 2 pairs of equal sides that are next to each other (adjacent).

A useful idea: a square is a special rectangle, and a rectangle is a special parallelogram. The properties build on each other, so the more special the shape, the more rules it follows.

Here is one of the most useful facts in geometry: the three angles inside any triangle always add up to $180\degree$. It does not matter if the triangle is big, small, thin or wide.

This means if you know two angles in a triangle, you can always find the third. If two angles are $50\degree$ and $70\degree$, the third is $180\degree - 50\degree - 70\degree = 60\degree$.

For a quadrilateral, the four angles always add up to $360\degree$. Why? Because any quadrilateral can be split into two triangles by drawing one diagonal — and $2 \times 180\degree = 360\degree$.

So if three angles of a quadrilateral are $80\degree$, $95\degree$ and $100\degree$, the fourth is $360\degree - 80\degree - 95\degree - 100\degree = 85\degree$.

These two facts let you find missing angles without measuring — just careful subtraction.

What about a pentagon, hexagon or even a 12-sided shape? You can find the total of all the inside (interior) angles using a clever trick: split the polygon into triangles.

Pick one corner and draw lines to every other corner. Each triangle contributes $180\degree$.

A polygon with $n$ sides always splits into $(n - 2)$ triangles. So the interior angle sum is:

$\text{angle sum} = (n - 2) \times 180\degree.$

• Triangle ($n = 3$): $(3 - 2) \times 180\degree = 180\degree$.
• Quadrilateral ($n = 4$): $(4 - 2) \times 180\degree = 360\degree$.
• Pentagon ($n = 5$): $(5 - 2) \times 180\degree = 540\degree$.
• Hexagon ($n = 6$): $(6 - 2) \times 180\degree = 720\degree$.

For a regular polygon, every interior angle is equal — so divide the total by the number of sides. A regular pentagon has each angle $540\degree \div 5 = 108\degree$.

Symmetry is a property that helps you describe and classify shapes neatly.

A line of symmetry is a fold line: if you folded the shape along it, the two halves would match exactly. A square has 4 lines of symmetry; a rectangle has 2; an equilateral triangle has 3.

Rotational symmetry is about turning. The order of rotational symmetry tells you how many times a shape looks the same as you spin it once all the way round. A square has rotational symmetry of order 4; a rectangle of order 2; a scalene triangle of order 1 (it only matches in its starting position).

Symmetry gives you another quick way to sort shapes. A regular polygon with $n$ sides always has $n$ lines of symmetry and rotational symmetry of order $n$ — neat and predictable.

Getting confident with shapes and polygons opens the door to lots of later geometry:

• Perimeter, area and volume — you need to recognise a shape before you can measure it.
• 3D solids and nets — every face of a cuboid or prism is a polygon you have already met.
• Angle reasoning — finding missing angles in more complex diagrams builds straight on the $180\degree$ and $360\degree$ facts.
• In everyday life — design, art, architecture, sports pitches and even video-game graphics all rely on knowing how shapes behave.

If a later geometry topic feels tricky, it is often a shape property underneath that needs a quick refresh. Come back to this guide whenever you need it.