Question

question 8 of 25
what kinds of regular polygons can be used for regular tessellations?
check all that apply.
a. five - sided
b. three - sided
c. six - sided
d. eight - sided

Explanation:

Step1: Formula for interior angle of a regular polygon

The formula for the interior angle of a regular polygon is $\theta=\frac{(n - 2)\times180^{\circ}}{n}$, where $n$ is the number of sides.

Step2: Condition for regular tessellation

For a regular polygon to be used in a regular tessellation, $\frac{360^{\circ}}{\theta}$ must be an integer.

• For \(n = 3\) (three - sided):
• Calculate the interior angle: $\theta=\frac{(3 - 2)\times180^{\circ}}{3}=60^{\circ}$.
• Check the tessellation condition: $\frac{360^{\circ}}{60^{\circ}} = 6$, which is an integer.
• For \(n = 5\) (five - sided):
• Calculate the interior angle: $\theta=\frac{(5 - 2)\times180^{\circ}}{5}=108^{\circ}$.
• Check the tessellation condition: $\frac{360^{\circ}}{108^{\circ}}=\frac{10}{3}\approx3.33$, which is not an integer.
• For \(n = 6\) (six - sided):
• Calculate the interior angle: $\theta=\frac{(6 - 2)\times180^{\circ}}{6}=120^{\circ}$.
• Check the tessellation condition: $\frac{360^{\circ}}{120^{\circ}} = 3$, which is an integer.
• For \(n = 8\) (eight - sided):
• Calculate the interior angle: $\theta=\frac{(8 - 2)\times180^{\circ}}{8}=135^{\circ}$.
• Check the tessellation condition: $\frac{360^{\circ}}{135^{\circ}}=\frac{8}{3}\approx2.67$, which is not an integer.

Answer:

B. Three - sided, C. Six - sided