Extend the sides of the hexagon to form an equilateral triangle (try to see why it is an equilateral triangle). Then the area of this triangle is sqrt(3)/4 * a^2 where a = a1 + a2 + a3 is the length of the side of the big triangle. Notice that the area of the original hexagon equals to the Area of big triangle - Area of 3 triangles ==> the area = sqrt(3)/4 * a^2 - sqrt(3)/4 * a1^2 - sqrt(3)/4 * a3^2 - sqrt(3)/4 * a5^2. where a1, a3, and a5 are the sides of the 3 triangles we added to the hexagon to form the big triangle.

Now to see how many triangles of side = 1 exists we divide the area of the hexagon by the area of an equilateral triangle of side length = 1 (which equals sqrt(3)/4) ==> the answer = a^2 - a1^2 - a3^2 - a5^2.