Question

question 4 of 10
if a tessellation is regular, how many sides can the tessellating regular
polygon have?
check all that apply.
a. 3
b. 10
c. 4
d. 5
e. 6
f. 9

Explanation:

Step1: Calculate interior angle formula

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: Check tessellation condition

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

• For \(n = 3\):

$\theta=\frac{(3 - 2)\times180^{\circ}}{3}=60^{\circ}$, and $\frac{360^{\circ}}{60^{\circ}} = 6$ (integer).

• For \(n = 4\):

$\theta=\frac{(4 - 2)\times180^{\circ}}{4}=90^{\circ}$, and $\frac{360^{\circ}}{90^{\circ}} = 4$ (integer).

• For \(n = 6\):

$\theta=\frac{(6 - 2)\times180^{\circ}}{6}=120^{\circ}$, and $\frac{360^{\circ}}{120^{\circ}} = 3$ (integer).

• For \(n = 5\):

$\theta=\frac{(5 - 2)\times180^{\circ}}{5}=108^{\circ}$, and $\frac{360^{\circ}}{108^{\circ}}=\frac{10}{3}$ (not integer).

• For \(n = 9\):

$\theta=\frac{(9 - 2)\times180^{\circ}}{9}=140^{\circ}$, and $\frac{360^{\circ}}{140^{\circ}}=\frac{18}{7}$ (not integer).

• For \(n = 10\):

$\theta=\frac{(10 - 2)\times180^{\circ}}{10}=144^{\circ}$, and $\frac{360^{\circ}}{144^{\circ}}=\frac{5}{2}$ (not integer).

Answer:

A. 3, C. 4, E. 6