Look at the picture.
The apothem is a height of an equilateral triangle. The formula of a height of an equilateral triangle is:
[tex]h=\dfrac{r\sqrt3}{2}[/tex]
We have [tex]h=6\sqrt3[/tex].
Substitute:
[tex]\dfrac{r\sqrt3}{2}=6\sqrt3[/tex] multiply both sides by 2
[tex]r\sqrt3=12\sqrt3[/tex] divide both sides by √3
[tex]\boxed{r=12}[/tex]
The exact length of the radius
of regular hexagon ABCDEF: 12 inches.
The exact length of the perimeter
of regular hexagon ABCDEF: 6·12 in = 72 in.
You can use the Pythagorean theorem:
[tex]r^2=\left(\dfrac{r}{2}\right)^2+(6\sqrt3)^2\\\\r^2=\dfrac{r^2}{4}+6^2(\sqrt3)^2\\\\\dfrac{4r^2}{4}=\dfrac{r^2}{4}+(36)(3)\qquad\text{subtract}\ \dfrac{r^2}{4}\ \text{from both sides}\\\\\dfrac{3r^2}{4}=108\qquad\text{multiply both sides by 4}\\\\3r^2=432\qquad\text{divide both sides by 3}\\\\r^2=144\to r=\sqrt{144}\\\\\boxed{r=12}[/tex]
