Symmetry · Lesson 4 of 7
Rotation and Symmetric Turns
“Follow a figure around a fixed centre and identify the turns that leave its appearance unchanged.”
• Identify the centre of rotation and a matching angle of rotation. • Relate quarter, half and three-quarter turns to degree measures. • Distinguish rotational symmetry from the full turn that every figure makes. • Track labelled vertices through a rotation. • Explain why a figure may match under rotation even when no mirror fold works.
A different way for a figure to match
A paper windmill can look balanced even though folding it down a central line does not make its blades overlap. The blades lean in the same turning direction, so a reflection changes their arrangement. Turning the complete windmill around its central pin can produce a match instead. This is a different movement from reflection.
Place a cut-out windmill over an identical drawing. Keep the drawing still and turn only the cut-out around its centre. The cut-out must remain in the plane of the paper: do not flip it over. Compare the boundary after a small turn, then after a quarter-turn. The first may fail even when the second matches perfectly.
The fixed point about which a figure turns. The point itself stays in place while the other points move around it.
An angle of rotation about a specified centre that makes the entire figure coincide with its original appearance. In this chapter we list positive matching angles through one full turn, including 360°.
A figure has rotational symmetry when it matches itself after a turn greater than 0° and less than 360° about a fixed centre. Returning after 360° alone does not establish rotational symmetry.
A quarter of a full turn is 90°, a half-turn is 180° and three quarters of a full turn is 270°. Every figure returns to its original position after a full turn of 360°. This universal return is useful in counting angles, but the interesting question is whether the figure returns sooner.
Problem
The windmill has four identical blades equally arranged around its pin. What are its angles of symmetry in one full turn?
- 1.A 90° clockwise turn moves every blade into the previous position of the next blade. Because their shapes and orientations match, the whole boundary coincides.
- 2.Another 90° gives a total turn of 180°, and a third gives 270°. These positions also match.
- 3.At 360° the windmill returns to its starting position. The complete positive list is 90°, 180°, 270°, 360°. A smaller arbitrary angle, such as 20°, does not align the blades.
A full turn is always a matching angle, even for an irregular figure. Saying “it matches after 360°” is therefore insufficient to claim rotational symmetry. Also, turn the entire picture around one fixed point; sliding it or flipping it is a different test.
Follow corners, not just the outline
A square makes the movement easy to record because its corners are distinct positions. Imagine a transparent square labelled A, B, C, D clockwise, starting at the upper-left corner. Keep a second, unlabelled square underneath it. Turn the transparent square around their common centre and record where the labels go.
Problem
Where does each corner go after a 90° clockwise turn of the displayed square?
- 1.The upper-left corner A moves to the upper-right corner’s position. Thus A occupies B’s original position.
- 2.B moves to C’s original lower-right position; C moves to D’s lower-left position; D moves to A’s upper-left position.
- 3.The new labels around the stationary corner positions are D, A, B, C. The outline still overlaps the original square even though none of its corners stayed at its starting position.
Problem
What corner movement occurs after a 180° clockwise turn of the square?
- 1.A half-turn consists of two quarter-turns. Follow A first from upper-left to upper-right, then to lower-right.
- 2.Thus A and C exchange positions, while B and D exchange positions. Each corner arrives at the opposite corner.
- 3.The outline matches again. A further quarter-turn gives the 270° position, and the next returns every labelled corner to its original place at 360°.
The square’s centre is the intersection of its diagonals. Rotating about a corner rather than the centre would move the square to a different place. A rotation question therefore needs both a centre and an angle; a matching angle about one point may fail about another.
A similar-looking turn may still fail
Some figures appear balanced because their left and right sides match, yet they do not return under a smaller rotation. A symmetric trapezium with a short top and long bottom is an example. Reflection across its vertical axis works, but a half-turn exchanges the short and long bases.
Problem
The displayed trapezium has a short upper base and a longer lower base. Does turning it 180° preserve it?
- 1.A half-turn sends the upper base to the lower position and the lower base to the upper position.
- 2.The new lower base would be short where the original was long, so the two full outlines do not coincide.
- 3.Only the 360° full turn returns this trapezium. It has no rotational symmetry, although its vertical mirror line gives reflection symmetry.
A rectangle with unequal adjacent sides behaves differently. A 90° turn puts its longer side along the direction of the shorter side and fails to match. A 180° turn pairs each side with the opposite side of the same length and does match. Do not use the square’s quarter-turn result for every four-sided figure.
Draw a nonsquare rectangle, a symmetric trapezium and a four-blade windmill. Trace a second copy of each, identify its proposed centre and use the cut-out as an overlay. Check 90° and 180°, then search for any earlier match. Record failed turns as well as successful ones; the failures explain the smallest successful turn.
Geometry and decoration must be distinguished
A plain square matches after a quarter-turn. A square with one red corner and three blue corners generally does not: the red corner moves to a blue position. If the question includes the colours, they are part of the figure. If labels or temporary colours only show which part moved in an explanation, test the geometric outline instead.
Problem
A square outline has a red dot at one corner and no dots at the other three. Does the entire decorated drawing have a 90° matching turn?
- 1.The outline alone matches after 90°, but the red dot moves to the next corner.
- 2.The original drawing has no red dot at that corner, so the decorated drawings differ.
- 3.The decoration returns only after 360°. This complete decorated figure therefore lacks rotational symmetry, despite the symmetry of its unadorned outline.
Quiz
Which turn is a quarter of a full turn?
Which observation proves rotational symmetry about a chosen centre?
In a 90° clockwise rotation of the labelled square, where does A go?
Why does the symmetric trapezium fail the 180° test?
What stays fixed during a rotation?
Which positive angle is a matching turn for every figure about its rotation point?
360° divided into four equal parts gives 90°.
Practice Problems
- Explain the difference between folding a windmill and turning it around its centre.
- List all matching angles of a plain square from greater than 0° through 360°. Track the corners at each one.
- List the matching angles of a nonsquare rectangle. Explain why 90° fails.
- Trace the symmetric trapezium and check the 180° mismatch using the unequal bases.
- Draw a square with a decoration that destroys its quarter-turn match but preserves its half-turn match.
- Explain why a 360° match does not distinguish a symmetric figure from an asymmetric one.
- Compare a rotation about a square’s centre with one about a corner. State what makes the latter fail to overlap the original.
- Give one figure with reflection symmetry but no smaller matching rotation, and one with a smaller matching rotation but no mirror line.
A plain square matches at 90°, 180°, 270°, 360°. A nonsquare rectangle matches at 180° and 360° only.
Key Takeaways
• A rotation turns a figure about one fixed centre. • A matching angle must preserve the entire figure under the specified test. • Every figure matches after 360°; rotational symmetry requires an earlier match. • A square matches at quarter-turn intervals, while a nonsquare rectangle matches at half-turn intervals. • Reflection symmetry and rotational symmetry are different properties. • Decoration can reduce the symmetry of an otherwise symmetric outline.