Answer:
[tex]A=54\sqrt{3}[/tex]
Step-by-step explanation:
here we are going to use the formula which is
Area=[tex]\frac{1}{2} \times P \times A[/tex]
Where P is perimeter and A is apothem
Please refer to the image attached with this :
In a Hexagon , there are six equilateral triangle being formed by the three diagonals which meet at point O.
Consider one of them , 0PQ with side a
As Apothem is the Altitude from point of intersection of diagonals to one of the side. Hence it divides the side in two equal parts . hence
[tex]PR = \frac{a}{2}[/tex]
Also OP= a
Using Pythagoras theorem ,
[tex]OP^2=PR^2+OR^2[/tex]
[tex]a^2=(\frac{a}{2})^2+(\3sqrt{3})^2[/tex]
[tex]a^2=\frac{a^2}{4}+27[/tex]
Subtracting both sides by [tex]\frac{a^2}{4}[/tex]
[tex]a^2-\frac{a^2}{4}=27[/tex]
[tex]\frac{4a^2-a^2}{4}=27[/tex]
[tex]\frac{3a^2}{4}=27[/tex]
[tex]a^2=\frac{4 \times 27}{3}[/tex]
[tex]a^2=4 \times 9[/tex]
[tex]a^2=36[/tex]
taking square roots on both sides we get
[tex]a=6[/tex]
Now we have one side as 6 mm
Hence the perimeter is
[tex]P=6 \times 6[/tex]
[tex]P=36[/tex] mm
Apothem = [tex]3\sqrt{3}[/tex]
Now we put them in the main formula
Area = [tex]\frac{1}{2} \times 36 \times 3\sqrt{3}[/tex]
Area=[tex]18 \times 3\sqrt{3}[/tex]
Area=[tex]54\sqrt{3}[/tex]
