SOLUTION: In hexagon ABCDEF, AB = DE = 2, BC = EF = 4$, CD = FA = 6, and all the interior angles are equal. Find the area of hexagon ABCDEF.

 

It's easy to show by the formula for the sum of the angles that
each interior angle is 120°, whose supplement is 60°.

Now we draw in some green lines to enclose the hexagon in a rectangle.
We have added four 30°-60°-90° right triangles to the hexagon.

 

In a 30°-60°-90° right triangle the shorter leg
is one-half of the hypotenuse.  And the longer leg
is the shorter leg times √3.

Since we know all 4 hypotenuses, we can find the dimensions
of the enclosing rectangle. 

 

The area of the big rectangle is 

(length)(width)= (9)(3√3) = 27√3

The area of each of the two larger 30°-60°-90° right triangles
is  

(base)(height)/2 = (2)(2√3)/2 = 2√3

The area of both of them is 4√3

The area of each of the two smaller 30°-60°-90° right triangles
is  

(base)(height)/2 = 1(√3)/2 = √3/2

The area of both of them is √3

We get the area of the hexagon by subtracting the
areas of the 4 right triangles:

27√3 - 4√3 - √3 = 22√3

Edwin