Hexagon Side and Circumscribing Circle Radius

A regular hexagon can be divided into 6 equilateral triangles, with the center of the hexagon being a common vertex. The side length of each equilateral triangle is equal to the side length of the hexagon, denoted as $a$. The radius ($r$) of the circle that circumscribes the hexagon is the distance from the center to any vertex.

In a regular hexagon, this distance (the radius $r$) is equal to the side length of the hexagon.

• Radius of the circumscribing circle, $r = a$.

Calculating Circle Area

The formula for the area ($A$) of a circle is given by:

$A = \pi r^2$

Substitute the radius $r = a$ into the area formula:

$A = \pi (a)^2$

$A = \pi a^2$

Therefore, the area of the circle that circumscribes a regular hexagon with side $a$ is $\pi a^2$.