Question

Regular hexagon ABCDEF is inscribed in circle X and has an apothem that is 6√3 inches long. Use the length of the apothem to calculate the exact length of the radius and the perimeter of regular hexagon ABCDEF. In your final answer, include your calculations.
The exact length of the radius of regular hexagon ABCDEF:
The exact length of the perimeter of regular hexagon ABCDEF:

Answer 1

Look at the picture.

The apothem is a height of an equilateral triangle. The formula of a height of an equilateral triangle is:

$h=\dfrac{r\sqrt3}{2}$

We have $h=6\sqrt3$.

Substitute:

$\dfrac{r\sqrt3}{2}=6\sqrt3$     multiply both sides by 2

$r\sqrt3=12\sqrt3$    divide both sides by √3

$\boxed{r=12}$

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:

$r^2=\left(\dfrac{r}{2}\right)^2+(6\sqrt3)^2\\\\r^2=\dfrac{r^2}{4}+6^2(\sqrt3)^2\\\\\dfrac{4r^2}{4}=\dfrac{r^2}{4}+(36)(3)\qquad\text{subtract}\ \dfrac{r^2}{4}\ \text{from both sides}\\\\\dfrac{3r^2}{4}=108\qquad\text{multiply both sides by 4}\\\\3r^2=432\qquad\text{divide both sides by 3}\\\\r^2=144\to r=\sqrt{144}\\\\\boxed{r=12}$