To find the area of a regular hexagon when we know the length of the apothem, we can use the formula:
Area = (Perimeter × Apothem) / 2
First, we need to find the perimeter. A regular hexagon is made up of 6 equal sides. The relationship between the apothem (a) and the side length (s) in a regular hexagon is given by:
a = (s × √3) / 3
Given the apothem is 6 m:
6 = (s × √3) / 3
Multiplying both sides by 3:
18 = s × √3
Now solve for s:
s = 18 / √3 = 6√3 m
Now we can calculate the perimeter:
Perimeter = 6 × s = 6 × 6√3 = 36√3 m
Now we have both the perimeter and the apothem. Let’s substitute these into the area formula:
Area = (Perimeter × Apothem) / 2 = (36√3 × 6) / 2
Simplifying this gives:
Area = (216√3) / 2 = 108√3
So, the area of the regular hexagon is 108√3 m² or approximately 187.94 m² when calculated numerically.