Question

If a regular hexagon circumscribes a circle of radius r, then its perimeter is

• A. $4r\sqrt{3}$
• B. $6\sqrt{3}r$
• C. $6r$
• D. $12\sqrt{3}r$

Answer 1

The correct option is A $4r\sqrt{3}$

Obviously, a regular hexagon can be considered as consisting of six equilateral triangles with one common vertex at the center of an inscribed circle.

The altitude of each of these triangles equals to $r$

The base of each of these triangles (a side of a hexagon that is perpendicular to an altitude-radius) equals to

$r=a\sin {60}^o\Rightarrow a=\frac{r}{\sin {60}^o}=\frac{2r}{\sqrt{3}}$

Hence, Perimeter $=6\ast \frac{2r}{\sqrt{3}}=4r\sqrt{3}$