Solution #9

 

Add point F as shown above, so that AF = BF = 2. Triangle ABF is then equilateral with side length 2.

Draw EC, and note that EC is parallel to AB, because angles EAB and ABC are equal. Thus angles AEC and ECB are both 60 degree angles, and triangles CDE and CEF are both equilateral with side length 4.

The area of the pentagon is then the area of the two equilateral triangles of side length 4, minus the area of the equilateral triangle of side length 2.

The area of an equilateral triangle is  s² √ 3  /4, where s is the side length. So our total area is 2*(4² √ 3  /4) - 2² √ 3  /4 = 7 √ 3  .