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Symmetry - Order of 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 looks exactly like its original position after being rotated about a fixed point by an angle less than 360∘360^{\circ}. For example, if you rotate a square by 90∘90^{\circ}, it will look identical to how it started.

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The fixed point around which the object is rotated is called the Center of Rotation. In a rectangle, the center of rotation is the point where the two diagonals intersect.

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The minimum angle through which a figure is turned to coincide with its original position is known as the Angle of Rotation. A full turn is 360∘360^{\circ}, a half-turn is 180∘180^{\circ}, and a quarter-turn is 90∘90^{\circ}.

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The Order of Rotational Symmetry is the number of times a figure fits onto itself during a complete rotation of 360∘360^{\circ}. If a shape looks the same at four different positions during a full turn, its order is 4.

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Every regular polygon with nn sides has an order of rotational symmetry equal to nn. For instance, a regular pentagon has an order of 5, meaning it looks the same at five positions: 72∘72^{\circ}, 144∘144^{\circ}, 216∘216^{\circ}, 288∘288^{\circ}, and 360∘360^{\circ}.

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A circle is a special case with infinite rotational symmetry. Because every point on the circumference is equidistant from the center, rotating a circle by any angle around its center results in a shape that looks exactly the same.

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It is important to note that every object has rotational symmetry of order 1, as it returns to its original position after a full 360∘360^{\circ} turn. However, in geometry, we usually say a figure has rotational symmetry only if its order is 2 or more.

📐Formulae

Order of Rotational Symmetry=frac360∘Angle of Rotation\text{Order of Rotational Symmetry} = \\frac{360^{\circ}}{\text{Angle of Rotation}}

Angle of Rotation=frac360∘Order of Rotational Symmetry\text{Angle of Rotation} = \\frac{360^{\circ}}{\text{Order of Rotational Symmetry}}

💡Examples

Problem 1:

Find the order of rotational symmetry for an equilateral triangle.

Solution:

  1. Identify the angle of rotation: For a regular polygon, the angle is determined by the symmetry of its sides. An equilateral triangle looks the same after every rotation of 120∘120^{\circ}.
  2. Use the formula: Order=frac360∘120∘Order = \\frac{360^{\circ}}{120^{\circ}}.
  3. Calculate: Order=3Order = 3.
  4. List the positions: The triangle coincides with itself at 120∘120^{\circ}, 240∘240^{\circ}, and 360∘360^{\circ}.

Explanation:

Since an equilateral triangle is a regular polygon with 3 equal sides, its order of rotational symmetry matches its number of sides.

Problem 2:

A ceiling fan has 5 identical blades. What is its angle of rotation and its order of rotational symmetry?

Solution:

  1. The fan has 5 identical blades arranged symmetrically around a center, making it act like a regular pentagon.
  2. The Order of Rotational Symmetry is equal to the number of blades: Order=5Order = 5.
  3. To find the Angle of Rotation, use the formula: Angle=frac360∘Order\text{Angle} = \\frac{360^{\circ}}{Order}.
  4. Calculate: Angle=frac360∘5=72∘\text{Angle} = \\frac{360^{\circ}}{5} = 72^{\circ}.

Explanation:

The order is determined by the number of repeating identical parts (the blades), and the angle is the full circle divided by that order.