Symmetry.
Find the reflections and rotations that map a plane shape or solid onto itself.
Identify line and rotational symmetry · describe symmetry planes and axes in solids
A reflection line splits a shape into matching halves
A line of symmetry is a fold line that makes the two halves of a shape coincide. The test is matching every point, not merely cutting the area into two equal amounts.
Imagine folding the paper or reflecting each boundary point across a candidate line. Points on the line remain fixed and corresponding points lie equally far from it. A line can be horizontal, vertical or slanted; the drawing’s orientation does not restrict possible symmetry lines.
A line of symmetry reflects the entire shape onto itself. Matching halves must coincide after folding; equal areas alone are not enough.
Pause and explainDoes a diagonal of every rectangle form a line of symmetry?
No. It does for a square, but a non-square rectangle’s halves do not coincide when folded along a diagonal.
Rotational order counts matches in one full turn
Rotational symmetry asks whether a shape fits its original outline again during a full turn about a centre. This is different from reflecting or folding it.
Count the number of matching positions in one complete 360° turn, including the original alignment once. The smallest positive matching angle is 360° divided by the order. A shape can have rotational symmetry even when it has no reflection symmetry.
The order is the number of times a shape fits exactly onto itself during 360°, including the final position. Order 1 means no smaller positive turn works.
| Shape | Reflection lines | Rotational order |
|---|---|---|
| Equilateral triangle | 3 | 3 |
| Isosceles triangle (not equilateral) | 1 | 1 |
| Scalene triangle | 0 | 1 |
| Square | 4 | 4 |
| Rectangle (not square) | 2 | 2 |
| Rhombus (not square) | 2 | 2 |
| Parallelogram (not rectangle or rhombus) | 0 | 2 |
| Kite (standard non-rhombus case) | 1 | 1 |
| Regular n-gon | n | n |
| Circle | Infinitely many | Any rotation |
Find the smallest angle that maps a regular hexagon onto itself.
It has six lines of reflection symmetry and rotational order 6.
Pause and explainA shape has rotational order 4. What is its smallest matching turn?
90°, because four equal turns complete 360°.
In three dimensions: planes and axes
For a solid, the analogue of a fold line is a plane of symmetry. A plane is a flat slice that divides the solid into two mirror-image parts.
Visualise the entire three-dimensional object, not just its front outline. A cut that splits the volume equally is not necessarily a symmetry plane. Check whether features on either side of the proposed plane correspond as reflected shapes.
A plane of symmetry divides a solid into mirror-image halves. An axis of rotational symmetry is a line about which the solid rotates to fit itself.
| Solid | Planes of symmetry | Principal rotational axis |
|---|---|---|
| Right regular n-gonal prism (not a cube) | n vertical planes and 1 halfway between the bases | Through base centres; order n |
| Right circular cylinder | Infinitely many through its axis, plus the mid-height plane | Through circular centres; any rotation |
| Right square-based pyramid | 4 planes through the apex | Through apex and base centre; order 4 |
| Right circular cone | Infinitely many planes through its axis | Through apex and base centre; any rotation |
A cylinder also has half-turn axes along diameters of its mid-height circle. A prism may have additional half-turn axes; the table names its principal axis. A cone has no horizontal reflection plane. A cube has 9 planes of symmetry; a cuboid with three distinct edge lengths has 3.
Pause and explainIs every cut through the centre of a cuboid a plane of symmetry?
No. The two parts must be mirror images of the whole solid’s features, which only certain planes achieve.
Your turn. Name the transformation.
Test each proposed line, turn or plane against the whole figure. Separate the symmetry type before counting it.
For rotational order, count matches over a full turn without double-counting the starting position at 360°. For line or plane symmetry, explain how corresponding parts coincide.
State the lines of symmetry and rotational order of a square.
Reveal worked answer
A shape has rotational order 5. Find its smallest positive rotational symmetry angle.
Reveal worked answer
How many planes of symmetry has a right square-based pyramid? Describe its main rotational axis.
Reveal worked answer
Does a general parallelogram have a line of symmetry?