Reflective Symmetry in 3D Shapes

Why this matters

Symmetry in 3D is everywhere in design and engineering: car bodies, aircraft, bottles and buildings are often built to be symmetrical, because symmetrical objects balance, look right and are easier to manufacture in matching halves.

From lines to planes

In 2D, a rectangle has two lines of symmetry. Stretch that rectangle back into a cuboid, and each line becomes a plane running through the shape — plus one more that cuts across the middle. That is why a cuboid has three planes of symmetry.

Counting planes for prisms

Prism (cross-section)	Lines in the cross-section	Planes of symmetry
Equilateral triangle	3	4
Isosceles triangle	1	2
Scalene triangle	0	1
Regular pentagon	5	6
Regular hexagon	6	7
Semicircle	1	2
Circle (a cylinder)	infinitely many	infinitely many

Notice that every prism has at least one plane of symmetry — the one halfway along.

The cube is a special case

A square prism that is not a cube has $4 + 1 = 5$ planes. But when the length matches the square’s side, the shape looks the same from every face, and you get extra diagonal planes. A cube has:

3 planes parallel to its faces, and
6 planes that slice diagonally through a pair of opposite edges,

making 9 in total.

Drawing tip. On isometric paper, shade one plane per copy of the shape. It is far clearer than trying to draw several planes on a single diagram.

Key rules and formulas

A plane of symmetry

A plane is a flat surface. A plane of symmetry cuts a 3D shape into two congruent halves that are mirror images of each other — the 3D version of a line of symmetry.

Cuboid and cube

A cuboid with three different edge lengths has 3 planes of symmetry, one parallel to each pair of faces. A cube has 9: those 3, plus 6 diagonal planes through opposite edges.

Prisms

$planes = (lines of symmetry of the cross-section) + 1$

Each line of symmetry of the cross-section gives a plane running the length of the prism, and there is always one more plane cutting the prism halfway along, parallel to its ends. (The cube is special: its length equals its width, which adds extra planes.)

Infinitely many

A cylinder has infinitely many planes through its axis, plus one across the middle. A sphere has infinitely many planes through its centre.

Worked examples

Worked example 1

A regular hexagonal prism

How many planes of symmetry does a regular hexagonal prism have?

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Checking a statement

Hannah says: "No prism can have exactly ten planes of symmetry." Is she right?

Try it yourself first, then reveal the solution one step at a time.

Prism	Lines of symmetry in cross-section	Planes of symmetry
Equilateral triangular prism	Row 1, Lines of symmetry in cross-section	Row 1, Planes of symmetry
Regular pentagonal prism	Row 2, Lines of symmetry in cross-section	Row 2, Planes of symmetry
Regular hexagonal prism	Row 3, Lines of symmetry in cross-section	Row 3, Planes of symmetry
Regular octagonal prism	Row 4, Lines of symmetry in cross-section	Row 4, Planes of symmetry

Prism

Lines of symmetry in cross-section

Planes of symmetry

Equilateral triangular prism

Regular pentagonal prism

Regular hexagonal prism

Regular octagonal prism