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Because the areas of square PRBA, RQDC and QPEF are 12, 10 and 17 respectively.

Therefore, PQ = √ 17, PR = √ 12, RQ = √ 10.

Because Helen's formula S=√[p(p-a)(p-b)(p-c)] and P in the formula is half circumference: p=(a+b+c)/2.

Therefore, the area of △PRQ is calculated as √455/4.

Because the triangle PQR, BCR, Dirk and AFP have the same area.

Therefore, the sum of the areas of triangle PQR, BCR, Dirk and AFP is √455.

So the area of hexagonal ABCDEF is:12+10+17+√ 455 = 39+√ 455.