helllp
1 Answer

Ok it would appear that you don’t have the side length of the hexagon which would have made it a lot easier as AREA = 1/2*perimeter*apothem or 3*sidelength*apothem.
With a regular hexagon if you draw lines from the centre to the vertices you will split the hexagon into 6 identical equilateral triangles. Then the next thing to note is that the height of the equilateral triangles ie from the midpoint of the hexagon to the point that bisects the base is equal to the apothem. ie the height of each equilateral triangle = 2*sqrt(3)
Bisect one of the equilateral triangles and we have a right angled triangle and
we can now use Pythagoras’s Theorem.
The square of the hypotenuse is equal to the sum of the squares of the other two sides.
Because we bisected an equilateral triangle where by definition all sides are of equal length, the hypotenuse of the right angle triangle is the side length of the equilateral triangle, one of the other sides of the right angle triangle is half the side length of the equilateral triangle, and the third side is the apothem.
Calling the side length of the equilateral triangle S and using Pythagoras’s theorem on the right angled triangle,
S^2 = (1/2S)^2 + (2sqrt 3)^2
S^2 = 1/4S^2 +2^2*(sqrt3)^2
S^2 = 1/4S^2 +4*3
S^2 = 1/4S^2 + 12
3/4S^2 = 12
S^2 = 12 *4/3
S^2 = 16
S = 4
Therefore the side of the equilateral triangle = 4
As the side of the hexagon = the side of the equilateral triangle, the side of the hexagon also = 4
The area of the hexagon = 3*side length * apothem
= 3 * 4 * 2sqrt 3
= 24 sqrt 3
=41.56922