Let's consider the regular hexagon shown below and let s be the length of the side.
Next, let's draw segments joining the vertices with the center of the hexagon, which divides it into 6 triangles. To find the area of the hexagon we find the area of a triangle and multiply it by 6.
A_(hexagon) = 6A_(△ CXY)
Now, let's take a close look at △ CXY. Notice that CY=CX since both are radii of the hexagon.
The Base Angles Theorem tells us that the base angles are congruent, and so m ∠ XYC = m ∠ YXC. Also, the central angle (∠ YCX) has a measure of 360^(∘)6 = 60^(∘).
To find the area of the equilateral triangle above, we draw one altitude that bisects ∠ YCX and XY.
By the 30^(∘)-60^(∘)-90^(∘) Triangle Theorem, we conclude that the length of the longer leg (CD) is sqrt(3) times the length of the shorter leg.
CD = sqrt(3)* s/2
Thus, the area of the equilateral triangle is given by the following expression.
A_(△ CXY) = s* sqrt(3)s/2/2
⇓
A_(△ CXY) = sqrt(3)s^2/4
In conclusion, because there are six triangles making up the hexagon, the following expression gives us the area of the hexagon.
A_(hexagon) = 6sqrt(3)s^2/4
⇓
A_(hexagon) = 3sqrt(3)s^2/2
