• The order of rotational symmetry is how many times a shape looks identical during one full 360° turn — and it can never be zero (minimum is 1).
• A line of symmetry is a fold line — fold the shape along it and the two halves land exactly on top of each other.
• You need to know the names and side-counts of 2D polygons from triangle (3 sides) up to decagon (10 sides).
• Triangles split into 4 types (equilateral, isosceles, right-angled, scalene) and quadrilaterals split into 6 types (square, rectangle, parallelogram, rhombus, trapezium, kite) — each with its own side/angle/diagonal/symmetry rules.
• Circles have their own vocabulary: circumference, diameter, radius, arc, sector, chord, segment, tangent.
• 3D shapes (cube, cuboid, cylinder, prisms, pyramids, cone, sphere, tetrahedron) each have a fixed number of faces, edges, and vertices you should memorise.
• A plane of symmetry is the 3D version of a line of symmetry — an imaginary flat "slice" that splits a solid into two mirror-image halves.
• 3D shapes can also have rotational symmetry about an axis — depending on which axis you spin around.

What is the "order" of rotational symmetry?

Imagine pinning a shape to a table by its exact centre and spinning it around, like a wheel. As you spin it through a full 360° turn, sometimes the shape will look exactly the same as it did at the start — as if you hadn't moved it at all. The order of rotational symmetry is simply a count of how many times this "looks the same" moment happens during one complete 360° rotation.

Here's the key insight that trips people up: getting back to your starting position always counts as one of those "looks the same" moments. That means every single shape, no matter how irregular, has an order of rotational symmetry of at least 1 — a shape can never have order 0. If a shape only matches itself once (back at the start, after the full spin), we say it has rotational symmetry of order 1, which in plain English just means "no rotational symmetry" — because matching only when you're back to the beginning isn't really symmetry at all.

Worked Example

A plus-sign (cross) shape made of 5 squares, with certain squares shaded, is rotated about its centre. Tracing paper shows the shape matches the original twice during one full 360° turn (once part-way through, and once back at the very start).

What is line symmetry?

Line symmetry (sometimes called reflective or mirror symmetry) is about adding an imaginary straight line — a mirror line — across a shape such that each side is a perfect reflection of the other. The clearest way to think about this is as a fold line: if you physically folded the shape along that line, the two halves would sit exactly on top of each other, edge for edge, corner for corner.

Working with diagonal lines of symmetry & "two-way" reflections

Some questions give you half a shape plus a line of symmetry and ask you to complete the other half. If the line of symmetry is diagonal, this gets trickier to do by eye — the recommended method is to use tracing paper: trace the given portion, flip the tracing paper over along the line of symmetry, and draw what appears.

Watch out for "two-way" reflections: these happen when the mirror line actually passes through the shape itself (not just alongside it). In that case, the part of the shape on the mirror line's side needs to be reflected too — it's easy to forget this second reflection and only complete half the picture.

Worked Example

A shape made of shaded and unshaded squares has exactly 1 line of symmetry (a vertical line through the centre). The question asks: shade exactly 4 more squares so the shape has 4 lines of symmetry instead.

Polygons by number of sides

A polygon is any flat (2D) shape made of n straight sides. A regular polygon is the special case where all sides are the same length AND all angles are the same size (an irregular polygon might have the right number of sides but mismatched lengths/angles).

Types of triangle

Types of quadrilateral

Every quadrilateral question is really a "how many boxes does it tick" question. The table below is the single most useful thing to memorise in this whole topic.

Circle vocabulary

Faces, edges, and vertices — the vocabulary

Every 3D (solid) shape can be described using three building blocks:

Prisms and the cross-section idea

A prism is any 3D shape that has the exact same cross-section (the same flat "slice" shape) running all the way through its length. Think of it like a loaf of bread — however thick a slice you cut, that slice always looks the same shape. A cube's cross-section is a square, a cuboid's is a rectangle, a triangular prism's is a triangle.

A cylinder works the same way, except its cross-section is a circle — so it's sometimes thought of as a "circular prism," even though it technically has a curved surface rather than flat rectangular ones.

Pyramids and cones

A pyramid has a flat base (which could be a square, rectangle, triangle, or any polygon) and sloping triangular sides that all meet up at a single point at the top, called the apex. A triangular-based pyramid has the special name tetrahedron — all four of its faces are triangles.

A cone is essentially a pyramid with a circular base instead of a polygon base.

Faces / Edges / Vertices reference table

What is a plane of symmetry?

A plane is just a flat surface — think of it as an imaginary flat "slice" you could push through a 3D shape, and that slice could be any 2D shape (a square, a rectangle, a circle, whatever fits). A plane of symmetry is a plane that cuts the 3D shape into two congruent (identical) halves that are mirror images of each other. This is genuinely just the 3D version of a line of symmetry — instead of folding a flat shape along a line, you're slicing a solid along a flat surface.

Why does a cube have 9, not just 3? Because a cube has two types of symmetry plane: 3 planes that slice straight across parallel to a pair of faces (like cutting a loaf of bread), and 6 more diagonal planes that slice through pairs of opposite edges. 3 + 6 = 9. A cuboid only has the first type (3 planes) because its diagonal "slices" don't produce matching halves unless two of its dimensions happen to be equal.

Worked Example

A cuboid has length 8 cm, width 5 cm and height 11 cm. How many planes of symmetry does it have?

Can 3D shapes have rotational symmetry?

Yes! Just like 2D shapes can spin around a central point, 3D shapes can spin around a line called an axis of rotation. The order of rotational symmetry depends entirely on which axis you choose to spin around — the same solid can have different orders for different axes.

• I can find the order of rotational symmetry of a 2D shape using tracing paper and an arrow.
• I understand why every shape has an order of rotational symmetry of at least 1 (never 0).
• I can draw all the lines of symmetry on a given 2D shape, including diagonal ones.
• I know a rectangle's diagonals are NOT lines of symmetry.
• I can complete a half-drawn shape given a line of symmetry, including "two-way" reflections.
• I know the names of polygons from 3 sides (triangle) to 10 sides (decagon).
• I know the 4 types of triangle and their side/angle properties.
• I know the properties (sides, angles, diagonals, symmetry) of all 6 quadrilateral types.
• I know the circle vocabulary: circumference, diameter, radius, arc, sector, chord, segment, tangent.
• I can state the number of faces, edges and vertices for cube, cuboid, cylinder, prisms, pyramids, and tetrahedron.
• I understand what a plane of symmetry is and can visualise it as a "slice" through a solid.
• I know the plane-of-symmetry counts for cube (9), cuboid (3), and cylinder (infinite).
• I can find the order of rotational symmetry of a 3D shape about a given axis.