A turn that lands the shape on itself
Turn a shape about its center. If, before a full turn is complete, it lands exactly on its own outline, so that it looks the same as when it started, the shape has rotational symmetry.
This shape is made of three identical triangles set evenly around the origin. Turn it a third of a turn, 120°, and each triangle moves into the place of the next one. The outline after the turn is the outline before it.
Three identical triangles around the center, the gold dot. A turn of 120° about the center carries each triangle onto the next.
The order of rotational symmetry
The order of rotational symmetry is the number of positions in one full turn in which the shape looks exactly as it did at the start. The position after the full turn counts, and the starting position is not counted a second time.
The three-bladed shape looks the same after 120°, after 240° and after 360°, so its order is 3. The landings come at equal steps, and the smallest step is 360° divided by the order: 360° ÷ 3 = 120°. Every other landing is a whole number of those steps.
Squares and rectangles
A square turned 90° about its center puts each corner where the next corner was, and each side where the next side was, because all four sides are the same length. It looks the same after 90°, 180°, 270° and 360°, so its order is 4, and 360° ÷ 4 = 90°.
A rectangle that is not a square has order 2. A turn of 180° puts each long side where the other long side was. A turn of 90° does not work: it stands the rectangle on a short side, so the long sides point up and down where the short sides were.
A square centered on the gold dot. Each quarter turn lands it on its own outline, so every copy coincides with the square itself: order 4.
Regular polygons
A regular polygon has all its sides the same length and all its angles the same size, and its corners are equally spaced around its center. A turn of 360° divided by the number of sides moves each corner to the next corner, so a regular polygon with n sides has order n.
An equilateral triangle has order 3, with a smallest angle of 120°. A regular pentagon has order 5 and a smallest angle of 360° ÷ 5 = 72°. A regular hexagon has order 6 and a smallest angle of 60°.
A regular pentagon. A turn of 72° about its center moves each corner to the next corner, five times in a full turn: order 5.
Order 1
Every shape looks the same after a full turn of 360°, so every shape has order at least 1. Order 1 means that is the only turn that works: the shape has no rotational symmetry. There is no order 0.
This trapezium has order 1. Turned 180°, its long side would be at the top, where the short side is, so it does not land on itself before the full turn.
The trapezium has rotational symmetry of order 1, and one line of symmetry, drawn dashed.
Rotational symmetry and line symmetry
A line of symmetry is a fold line, as in Symmetry; rotational symmetry is a turn. They are different properties, and a shape can have either one without the other. The trapezium has a line of symmetry but order 1. A pinwheel of four identical blades has order 4 but no line of symmetry: its blades all lean the same way, so any mirror image of it would lean the other way. A parallelogram that is not a rectangle has order 2 and no line of symmetry.
A regular polygon has both: n lines of symmetry and order n. That is no coincidence. Two reflections in lines that cross at an angle make a rotation through twice that angle, as in Combined Transformations. A square's lines of symmetry are 45° apart, and 2 × 45° = 90°, the square's smallest turn. A rectangle's two lines are 90° apart, and 2 × 90° = 180°, its half turn. So any shape with two lines of symmetry also has rotational symmetry.
A pinwheel of four identical right triangles around the gold center. A quarter turn carries each blade onto the next, so its order is 4, but no fold line matches its two halves.
The usual mistakes
Treating a rectangle as a square. A rectangle that is not a square has order 2, not 4: a quarter turn stands it on its short side.
Giving order 0 to a shape with no rotational symmetry. The full turn always works, so the least order is 1.
Counting lines of symmetry instead of turns. The trapezium has one line of symmetry, but its order is 1 because of turns, not folds; the pinwheel has no line of symmetry and order 4.
Counting corners and sides together. An equilateral triangle has order 3, not 6: only the three turns of 120°, 240° and 360° land it on itself.
Worked example: The Order of Rotational Symmetry of a Regular Pentagon and of a Rectangle
Question (a) What is the order of rotational symmetry of a regular pentagon? (b) What is the order of rotational symmetry of a rectangle that is not a square?
1.A regular pentagon has 5 equal sides and 5 equal angles. Turn it about its center by 360° ÷ 5 = 72°: each corner moves to where the next corner was, so the pentagon fits its outline.
Turn the pentagon about its center by 360° ÷ 5 = 72°: each corner moves to where the next corner was. 2.It fits again after 2 × 72°, 3 × 72°, 4 × 72° and 5 × 72° = 360°, the full turn. That is 5 times in one full turn.
It fits its outline at 72°, 144°, 216°, 288° and 360°: 5 times in one full turn. 3.(a) The order of rotational symmetry of a regular pentagon is 5.
(a) The order of rotational symmetry of the regular pentagon is 5. 4.Turn the rectangle by a quarter turn, 90°. Its long sides now stand where its short sides were, so it does not fit its outline.
After a quarter turn, the rectangle's long sides stand where its short sides were, so it does not fit. 5.(b) After a half turn, 180°, the rectangle fits its outline, and after the full turn it fits again. That is 2 times, so the order of rotational symmetry is 2.
(b) The rectangle fits its outline at 180° and at 360° only, so its order is 2.
Answer: (a) 5; (b) 2
Common mistakes
- Giving the rectangle order 4, as for a square. After a quarter turn the rectangle's long sides stand where its short sides were, so it does not fit. It fits only after a half turn and after the full turn, so its order is 2.
- Leaving out the fit at the end of the turn and giving the rectangle order 1. The rectangle fits its outline after a half turn and again after the full turn, when it is back where it started. Both fits count, so the order is 2.
Worked example: A Hubcap with Five Spokes, Lined Up Again for a Photograph
Question A car's hubcap has five identical spokes, equally spaced round its center, and no other markings. (a) Find the order of rotational symmetry of the hubcap, and the smallest angle through which it can be turned about its center to look exactly the same. (b) In a photo shoot, the car is rolled forward and the wheel turns through 1000°. Through what smallest further angle, in the same direction, must the wheel turn for the hubcap to look exactly as it did at the start?
1.As the hubcap turns about its center, every spoke moves into the place of the next one after a fifth of a turn. In one full turn this happens 5 times, the last time with every spoke back in its own place.
Every spoke moves into the place of the next one after a fifth of a turn, and this happens 5 times in a full turn. 2.(a) The order of rotational symmetry is 5, and the smallest angle is 360° ÷ 5 = 72°.
(a) The order of rotational symmetry is 5, and the smallest angle is 360° ÷ 5 = 72°. 3.The hubcap looks as it did at the start after any turn that is a whole-number multiple of 72°. Dividing, 1000 ÷ 72 is 13 remainder 64, because 13 × 72 = 936 and 1000 − 936 = 64.
1000 ÷ 72 is 13 remainder 64, since 13 × 72 = 936. 4.So after 1000° the wheel is 64° past the position it reached after 936°, which looked like the start. The next position that looks like the start comes after 14 × 72 = 1008°.
After 1000° the wheel is 64° past the match at 936°; the next match is at 14 × 72 = 1008°. 5.(b) The wheel must turn a further 1008 − 1000 = 8°. Check: 1008 ÷ 72 = 14 exactly, so after 1008° every spoke is where a spoke was at the start.
(b) The wheel must turn a further 1008 − 1000 = 8°.
Answer: (a) order 5, and the smallest angle is 72°; (b) 8°
Common mistakes
- Working with whole turns: 1000 − 2 × 360 = 280, and 360 − 280 = 80° further. That brings each spoke back to its own place, but the hubcap already looks the same whenever each spoke is where another spoke was, every 72°.
- Giving the remainder, 64°, as the answer. The remainder is how far the wheel has gone past the last position that looked like the start; the further turn is what is left to the next one, 72 − 64 = 8°.