Answer:
24√3 square units
Step-by-step explanation:
The side length of a regular hexagon is 2/√3 times the length of the apothem. For an apothem of 2√3, the side length is ...
s = (2/√3)(2√3) = 4
The perimeter P of the hexagon is 6 times the side length, so is 6×4 = 24.
The area is given by ...
A = 1/2Pa
where P is the perimeter and 'a' is the apothem.
A = 1/2(24)(2√3) = 24√3
The area of the hexagon is 24√3 square units.
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A hexagon can be divided into 6 congruent equilateral triangles. The apothem will be the altitude of one of those, so will divide the triangle into two triangles with angles 30°, 60°, and 90°. The ratios of the side lengths of such a triangle is something you might want to remember: 1 : √3 : 2. The altitude corresponds to the "√3" side, and the side length of the equilateral triangle corresponds to the "2" side. That is, we have the proportion ...
apothem : hexagon side = √3 : 2
Multiplying by 2 gives the actual values in the hexagon of this problem:
apothem : hexagon side = 2√3 : 4
