To find the area of the hexagon, divide it into triangles by drawing diagonals.
About 6.45 units
Practice makes perfect
To determine the side of the square, we first have to find the area of the hexagon.
Area of the Hexagon
Let's draw a regular hexagon including the diagonals between opposite sides. Notice that the diagonals bisect each other.
Since each of the six triangles are congruent and isosceles, their vertex angle is 360^(∘) divided by 6.
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.
