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Symmetry - Rotational Symmetry

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Rotational Symmetry occurs when an object or shape looks exactly the same as its original position after being rotated about a fixed point by a certain angle less than 360∘360^{\circ}. Visualize a ceiling fan or a windmill; as they spin, they appear identical at multiple positions before completing a full turn.

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The Center of Rotation is the fixed point around which the rotation takes place. In geometric figures, this is often the centroid or the intersection of diagonals. Imagine a pin pushed through the center of a cardboard square; the point where the pin stays still while the square turns is the center of rotation.

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The Angle of Rotation is the smallest angle by which a figure must be turned to look exactly like the original shape. For a square, this angle is 90∘90^{\circ} because rotating it by 90∘90^{\circ}, 180∘180^{\circ}, or 270∘270^{\circ} makes it occupy the same space and look identical.

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The Order of Rotational Symmetry is the total number of times a figure fits onto itself during a full 360∘360^{\circ} rotation. If a shape looks identical 44 times in one full circle (like a square), its order of rotational symmetry is 44. Every shape has at least an order of 11, as it will always look like itself after a full 360∘360^{\circ} turn.

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Regular Polygons have a special property where the order of rotational symmetry is equal to the number of sides. For instance, a regular hexagon has 66 sides, so it fits onto itself 66 times during a full rotation, giving it an order of 66.

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Line Symmetry vs. Rotational Symmetry: A figure can have both types of symmetry, only one, or neither. For example, the letter 'S' has rotational symmetry of order 22 about its center but has no lines of symmetry. A circle is the most symmetrical shape, possessing infinite lines of symmetry and an infinite order of rotational symmetry.

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Direction of Rotation can be clockwise or counter-clockwise. By convention, if the direction is not specified, rotation is often considered counter-clockwise in coordinate geometry, though for basic rotational symmetry, the order remains the same regardless of direction.

📐Formulae

Order of Rotational Symmetry=360∘Angle of Rotation\text{Order of Rotational Symmetry} = \frac{360^{\circ}}{\text{Angle of Rotation}}

Angle of Rotation=360∘Order of Rotational Symmetry\text{Angle of Rotation} = \frac{360^{\circ}}{\text{Order of Rotational Symmetry}}

For a Regular Polygon with n sides: Order=n\text{For a Regular Polygon with } n \text{ sides: Order} = n

Angle of Rotation for Regular Polygon=360∘n\text{Angle of Rotation for Regular Polygon} = \frac{360^{\circ}}{n}

💡Examples

Problem 1:

Determine the angle of rotation and the order of rotational symmetry for a regular octagon.

Solution:

  1. A regular octagon has n=8n = 8 equal sides.
  2. The order of rotational symmetry for a regular polygon is equal to the number of sides. Therefore, Order = 88.
  3. To find the angle of rotation, use the formula: Angle=360∘n\text{Angle} = \frac{360^{\circ}}{n}.
  4. Substitute n=8n = 8: Angle=360∘8=45∘\text{Angle} = \frac{360^{\circ}}{8} = 45^{\circ}.

Explanation:

Since all sides and angles of a regular octagon are equal, it will look identical every time it is rotated by the angle formed at its center by one side, which is 360∘360^{\circ} divided by 88.

Problem 2:

An equilateral triangle is rotated about its center. At what angles between 0∘0^{\circ} and 360∘360^{\circ} will the triangle look exactly like its starting position?

Solution:

  1. For an equilateral triangle, the number of sides n=3n = 3.
  2. The order of rotational symmetry is 33.
  3. The smallest angle of rotation is 360∘3=120∘\frac{360^{\circ}}{3} = 120^{\circ}.
  4. The triangle will look identical at multiples of this angle: 120∘×1=120∘120^{\circ} \times 1 = 120^{\circ}, 120∘×2=240∘120^{\circ} \times 2 = 240^{\circ}, and 120∘×3=360∘120^{\circ} \times 3 = 360^{\circ}.

Explanation:

The triangle repeats its appearance every 120∘120^{\circ}. The question asks for angles between 0∘0^{\circ} and 360∘360^{\circ}, so the positions are reached at 120∘120^{\circ} and 240∘240^{\circ} (and finally at 360∘360^{\circ} to complete the cycle).