The apothem of a regular polygon is a segment drawn the center of the polygon (that is, the center of the circle which circumscribes the polygon) to one side, such that it is perpendicular to the side.
Consider the regular hexagon inscribed in a circle , and are radii from the center of the circle to the two vertices of the hexagon. is drawn from the center of the regular polygon perpendicular to the side of the polygon. So, is an apothem.
Be careful with this formula, though: for a given apothem, there is only ONE POSSIBLE value of . (You can actually calculate , but it takes some work.)
For a regular hexagon with apothem m, the side length is about m.
Use the formula
to find the area of the hexagon.
The perimeter of the hexagon is about or m.
Now substitute the values.
Therefore, the area of the regular hexagon is about .