This MCQ module is based on: Rotational Symmetry and Angle of Rotation
TOPIC 34 OF 41
Rotational Symmetry and Angle of Rotation
🎓 Class 6
Mathematics
CBSE
Theory
Ch 9 — Symmetry
⏱ ~30 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Rotational Symmetry and Angle of Rotation
Targeting Class 6 level in Geometry, with Basic difficulty.
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9.4 Rotational Symmetry
Hold a pinwheel and blow on it — it spins but looks the same at certain positions. A flower, a wheel, a fan, a chakra — all show the same kind of symmetry: they appear unchanged after a turn. This is called rotational symmetry?.
Definition
A figure has rotational symmetry if it looks exactly the same after being rotated (turned) by some angle about a fixed point called the centre of rotation. The smallest such angle is called the angle of symmetry.The Centre of Rotation
Every rotation has a fixed point that does not move — the centre. For a pinwheel, the centre is the pin. For a square, it is where the two diagonals cross.
Example — A Square
Take a square ABCD. Mark the centre O (intersection of diagonals). Rotate the square about O by 90°. Where does each vertex go?
A square returns to an identical-looking position four times in a full turn.
Order of Rotational Symmetry
The order of rotational symmetry is the number of times a figure looks the same in one complete turn (360°). For a square, order = 4. For an equilateral triangle, order = 3. For a circle, order is infinite.Relation: angle of symmetry = \(\dfrac{360°}{\text{order}}\)
Trivial Rotation
Every figure looks the same after a 360° turn — this is not counted as rotational symmetry. So if a figure only matches itself at 360°, we say it has no rotational symmetry (order 1).
Angles and Orders — Common Figures
| Figure | Order | Angle of Symmetry | Centre of Rotation |
|---|---|---|---|
| Equilateral triangle | 3 | 120° | Centroid |
| Square | 4 | 90° | Intersection of diagonals |
| Rectangle (non-square) | 2 | 180° | Intersection of diagonals |
| Regular pentagon | 5 | 72° | Centre |
| Regular hexagon | 6 | 60° | Centre |
| Circle | infinite | any angle | Centre of the circle |
| Scalene triangle | 1 (none) | — | — |
Example — Rectangle (not square)
A rectangle ABCD (AB ≠ BC) rotated about the centre by 90° does not look the same — the longer and shorter sides swap places. But after 180°, it looks identical. So a non-square rectangle has order 2.
Non-square rectangle: order 2, angle of symmetry = 180°.
The Windmill
An NCERT pinwheel with 4 blades rotates by 90° to come back to a look-alike position. Its order is 4. A 3-bladed pinwheel has order 3 and angle 120°.
🔵 In-text question: Do you know of any shape that has exactly 4 angles of rotational symmetry? Yes — a square (angles 90°, 180°, 270°, 360° — but 360° is trivial, so 3 proper angles; the order is 4 counting the trivial rotation).
Activity: Trace-and-Turn
L3 ApplyMaterials: Two identical copies of a shape on tracing paper, pin/pencil tip, protractor
Predict: Of these four shapes — equilateral triangle, rectangle (non-square), letter S, regular hexagon — which do you think has the highest order of rotational symmetry?
- Draw each shape on a sheet of paper.
- Trace a copy of the shape on tracing paper.
- Pin the tracing at its centre on top of the drawing.
- Slowly rotate the tracing. Count how many times (before reaching 360°) the tracing lies exactly on the drawing. This count is the order.
- Compute the angle of symmetry = 360° ÷ order.
Equilateral triangle → 3 (angle 120°). Rectangle → 2 (angle 180°). Letter S → 2 (angle 180°). Regular hexagon → 6 (angle 60°). Hexagon has the highest order.
Figure it Out — Part B
Q1. State the order of rotational symmetry and angle of symmetry for: (i) equilateral triangle (ii) square (iii) regular hexagon (iv) scalene triangle (v) letter Z (vi) letter O.
(i) 3, 120° (ii) 4, 90° (iii) 6, 60° (iv) 1 (no rotational symmetry) (v) 2, 180° (vi) infinite (treating O as a circle).
Q2. Which of these angles can be an angle of rotational symmetry for some figure: 30°, 45°, 60°, 70°, 90°, 100°, 120°? Justify.
Valid angles must divide 360° exactly. 360/30=12 ✓, 360/45=8 ✓, 360/60=6 ✓, 360/70 ≠ integer ✗, 360/90=4 ✓, 360/100 ≠ integer ✗, 360/120=3 ✓. So 30°, 45°, 60°, 90°, 120° are valid.
Q3. Where is the centre of rotation of (a) a square, (b) a circle, (c) an equilateral triangle?
(a) Intersection of the two diagonals. (b) Centre of the circle. (c) Centroid — intersection of the three medians / three lines of symmetry.
Competency-Based Questions
Scenario: A company logo is made of a regular 5-pointed star (pentagram) surrounded by a circular band with the company name repeated 5 times, evenly spaced.
Q1. What is the order and angle of rotational symmetry of the 5-pointed star alone?
L3 ApplyOrder 5, angle 72°. The star looks the same after every turn of 72° (= 360°/5).
Q2. Does the entire logo (star + repeated text) still have rotational symmetry? If yes, what order? Analyse.
L4 AnalyseYes, order 5. The text repeats 5 times evenly (72° apart), matching the star's symmetry. So rotating the whole logo by 72° makes both the star and the text ring land in look-alike positions.
Q3. Suppose the designer writes the company name only once at the top of the circular band. Evaluate how that change affects the logo's rotational symmetry.
L5 EvaluateRotational symmetry is lost. The single text label breaks the 5-fold rotational pattern. The logo would now have only order 1 — no rotational symmetry — unless the text is itself symmetric (like "OXO") and placed on an axis of the star.
Q4. Create a sketch for a school emblem that has rotational symmetry of order 6 and also 6 lines of symmetry. What base shape would you start with?
L6 CreateStart with a regular hexagon. It already has order 6 and 6 lines of symmetry. Decorate it by placing an identical small motif inside each of the 6 triangles meeting at the centre — keeping every motif the same preserves both symmetries.
Assertion–Reason Questions
Assertion (A): A figure that has no line of symmetry can still have rotational symmetry.
Reason (R): Line and rotational symmetry are independent properties.
Reason (R): Line and rotational symmetry are independent properties.
(a) Example: the letter S or the number 6-9 motif — no line symmetry but 180° rotational symmetry. R is the correct reason.
Assertion (A): The order of rotational symmetry of a scalene triangle is 1.
Reason (R): A scalene triangle has three different side lengths, so no non-trivial rotation keeps it unchanged.
Reason (R): A scalene triangle has three different side lengths, so no non-trivial rotation keeps it unchanged.
(a) Only the trivial 360° rotation returns it to itself, because unequal sides cannot map onto each other. R explains A.
Assertion (A): Angle of rotational symmetry of a figure with order 5 is 75°.
Reason (R): Angle of symmetry = 360° ÷ order.
Reason (R): Angle of symmetry = 360° ÷ order.
(d) A is false — correct angle = 360°/5 = 72°, not 75°. R is the correct formula.
Frequently Asked Questions
What is the order of rotational symmetry?
The order of rotational symmetry is the number of times a figure looks identical to its original position during one complete 360-degree rotation.
What is the angle of rotation for a square?
A square has rotational symmetry of order 4. Its angle of rotation is 360/4 = 90 degrees.
What is the order of rotation of an equilateral triangle?
An equilateral triangle has rotational symmetry of order 3, with angle of rotation 120 degrees about its centroid.
Does every figure have rotational symmetry?
Every figure has trivial rotational symmetry of order 1 (a full 360-degree turn). We say a figure has rotational symmetry only if the order is 2 or more.
What is the centre of rotation?
The centre of rotation is the fixed point about which a figure is turned. For regular polygons and circles it lies at the centre of the shape.
Does a parallelogram have rotational symmetry?
A general parallelogram has rotational symmetry of order 2 about its centre (half-turn symmetry) but no line symmetry.
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Mathematics Class 6 — Ganita Prakash
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