In a figure, is the smallest angle of symmetry. What are the other angles of symmetry of this figure?

Question

Homi · Accepted Answer

Understand the Question

If a figure has a smallest angle of rotational symmetry $\theta$ , all successive rotations that return the figure to an identical orientation will be positive integer multiples of $\theta$ up to a full turn ( $360^\circ$ ).
Given the smallest angle of symmetry is $60^\circ$ , we find the remaining angles of symmetry by computing successive multiples of $60^\circ$ up to $360^\circ$ .

Step 1 · Find the Multiples of the Smallest Angle

Given that the smallest angle of symmetry is $60^\circ$ . Diagram 1

The other angles of rotational symmetry are the multiples of $60^\circ$ up to $360^\circ$ :

\begin{aligned} 60^\circ \times 2 &= 120^\circ \\ 60^\circ \times 3 &= 180^\circ \\ 60^\circ \times 4 &= 240^\circ \\ 60^\circ \times 5 &= 300^\circ \\ 60^\circ \times 6 &= 360^\circ \end{aligned}

Answer

$120^\circ, 180^\circ, 240^\circ, 300^\circ, \text{ and } 360^\circ$

Common Mistakes

Omitting $360^\circ$ : Forgetting that a full turn ( $360^\circ$ ) always brings every figure back to its original position, making it a valid angle of rotational symmetry.
Including the Smallest Angle: Listing $60^\circ$ again when the question specifically asks for the other angles of symmetry.

Question 16