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Identify rotational symmetry

Rotational symmetry is the property of a plane figure matching itself after a rotation about a fixed center through an angle less than 360°, with the figure’s position and appearance unchanged. A figure’s order of rotational symmetry is the number of times it coincides with itself during one full turn, and the smallest angle of symmetry is 360° divided by that order; this differs from reflection (line) symmetry.

Detailed Explanation: Identify rotational symmetry

Rotational symmetry means a figure matches itself when it is turned around a fixed center by an angle less than 360∘360^\circ. The figure must have the same position and appearance after the turn.

To identify it:

  1. Find the center of the figure.
  2. Imagine turning the figure around that center.
  3. Check whether it matches itself before a full turn.
  4. Count how many times it matches during one full turn. This is the order of rotational symmetry.
  5. Find the smallest angle using
smallest angle=360∘order.\text{smallest angle}=\frac{360^\circ}{\text{order}}.

Example

A parallelogram is shown. Does it have rotational symmetry? If so, find its order and smallest angle.

  1. The center is where the diagonals cross.
  2. Imagine rotating the parallelogram 180∘180^\circ around this center.
  3. Each vertex moves to the opposite vertex. The parallelogram matches its original position.
  4. A smaller turn, such as 90∘90^\circ, does not make it match.
  5. During one full turn, it matches itself twice: at 180∘180^\circ and again at 360∘360^\circ.

Therefore, the parallelogram has rotational symmetry of order 22.

Its smallest angle is

360∘2=180∘.\frac{360^\circ}{2}=180^\circ.

So, the answer is: order 22 and smallest angle 180∘180^\circ. This is different from reflection symmetry, which involves flipping a figure across a line.

Learn by doing: Identify rotational symmetry

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    2D Shape (Picture) Rotational Symmetry - How Many Orders of Symmetry (Not Shown)

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