360^(∘)/6=60^(∘)
If we draw the height from the vertex angle of one triangle, it will bisect the vertex angle and triangle's base.
In the diagram we see a right triangle. Since one of the non-right angles is 60^(∘), this is a 30^(∘)-60^(∘)-90^(∘) triangle. In such a triangle, the legs and hypotenuse have the following relationship.
The shorter leg in our triangle is 2 units. With this information, we can find the height.
h=asqrt(3) → h= 2sqrt(3)
Now we can calculate the area of the triangle and finally the hexagon's area by multiplying this number by 6.
Area: (1/2(4)2sqrt(3))6= 24sqrt(3) units^2
Side of the Square
When we know the area of the hexagon we also know the area of the square. Since a square's area is calculated by squaring its side, s, we can write the following equation.
s^2=24sqrt(3)
By solving for s we can determine the side of the square.
