Question

a regular polygon has possible angles of rotational symmetry of 20°, 40°, and 80°. how many sides does the polygon have?

10
12
18
20

Explanation:

Identify the rotational symmetry rule

For a regular polygon with \(n\) sides, the minimum angle of rotational symmetry is given by:

$$ \theta_{\text{min}} = \frac{360^\circ}{n} $$

Any other angle of rotational symmetry must be an integer multiple of this minimum angle:

$$ \theta = k \cdot \theta_{\text{min}} = k \cdot \frac{360^\circ}{n} $$

where \(k\) is an integer.

Analyze the given angles

We are given that the polygon has possible angles of rotational symmetry of \(20^\circ\), \(40^\circ\), and \(80^\circ\).
This means that \(20^\circ\), \(40^\circ\), and \(80^\circ\) must all be integer multiples of the minimum angle of rotational symmetry \(\theta_{\text{min}}\).
Thus, \(\theta_{\text{min}}\) must be a common divisor of \(20^\circ\), \(40^\circ\), and \(80^\circ\).
The greatest common divisor of \(20^\circ\), \(40^\circ\), and \(80^\circ\) is \(20^\circ\).
Therefore, the minimum angle of rotational symmetry \(\theta_{\text{min}}\) can be \(20^\circ\) (or a divisor of \(20^\circ\)).

Calculate the number of sides

Using the formula for the minimum angle of rotational symmetry:

$$ \theta_{\text{min}} = \frac{360^\circ}{n} $$

If \(\theta_{\text{min}} = 20^\circ\):

$$ 20^\circ = \frac{360^\circ}{n} $$

Solving for \(n\):

$$ n = \frac{360^\circ}{20^\circ} = 18 $$

Verify the options

Let's check if \(n = 18\) satisfies the conditions:

• If \(n = 18\), the minimum angle of rotational symmetry is \(\frac{360^\circ}{18} = 20^\circ\).
• The multiples of \(20^\circ\) are \(20^\circ\), \(40^\circ\), \(60^\circ\), \(80^\circ\), etc.
• This set of multiples contains \(20^\circ\), \(40^\circ\), and \(80^\circ\).
• Thus, a regular polygon with \(18\) sides has rotational symmetry at these angles.
• This matches the option \(18\).

Answer:

• 10
• 12
• 18 (Correct answer)
• 20