Using the properties of a 30-60-90 right triangle, you can readily determine that the measure of the apothem and the radius of the circumcircle (which is equal to the measure of a side of a regular hexagon) are in proportion:
Hence a regular hexagon with an apothem of 4 has a side that measures .
The perimeter of such a hexagon is simply 6 times the measure of a side, and therefore half of the perimeter is 3 times the measure of a side.
The area of a hexagon with perimeter and apothem is given by:
