Question

a regular hexagon is shown.
what is the measure of the radius, c, rounded to the nearest inch? use the appropriate trigonometric ratio to solve.
○ 6 in.
○ 10 in.
○ 14 in.
○ 24 in.

Explanation:

Step1: Identify hexagon central angle

A regular hexagon can be divided into 6 equilateral triangles, so the central angle for half of a triangle is $\frac{360^\circ}{12}=30^\circ$.

Step2: Relate to right triangle

The 12 in side is adjacent to the $30^\circ$ angle, and $c$ is the hypotenuse (radius). Use cosine: $\cos(30^\circ)=\frac{12}{c}$

Step3: Solve for $c$

Rearrange to isolate $c$: $c=\frac{12}{\cos(30^\circ)}$. Since $\cos(30^\circ)=\frac{\sqrt{3}}{2}\approx0.8660$, substitute:
$c=\frac{12}{0.8660}\approx13.856$

Step4: Round to nearest inch

$13.856\approx14$

Answer:

14 in.