Question

select the correct answer. which figure will tessellate the plane? a. regular pentagon b. regular decagon c. regular octagon d. regular hexagon

Explanation:

Step1: Recall the angle - sum formula for polygons

The sum of interior angles of a polygon with $n$ sides is given by $(n - 2)\times180^{\circ}$. So the measure of each interior angle $\theta$ of a regular polygon with $n$ sides is $\theta=\frac{(n - 2)\times180^{\circ}}{n}$.

Step2: Calculate interior angles for each option

For a regular pentagon ($n = 5$), $\theta=\frac{(5 - 2)\times180^{\circ}}{5}=108^{\circ}$. Since $360^{\circ}\div108^{\circ}=\frac{10}{3}$, not a whole - number, it does not tessellate.
For a regular decagon ($n = 10$), $\theta=\frac{(10 - 2)\times180^{\circ}}{10}=144^{\circ}$. Since $360^{\circ}\div144^{\circ}=\frac{5}{2}$, not a whole - number, it does not tessellate.
For a regular octagon ($n = 8$), $\theta=\frac{(8 - 2)\times180^{\circ}}{8}=135^{\circ}$. Since $360^{\circ}\div135^{\circ}=\frac{8}{3}$, not a whole - number, it does not tessellate.
For a regular hexagon ($n = 6$), $\theta=\frac{(6 - 2)\times180^{\circ}}{6}=120^{\circ}$. Since $360^{\circ}\div120^{\circ}=3$, it tessellates.

Answer:

D. regular hexagon