We can use something known as Heron's Formula to solve this
First.....find the semi-perimeter, S = [ 13 + 17 + 12sqrt (3) ] / 2 = 15 + 6sqrt (3)
Area = sqrt [ S * ( S -A) * (S - B) * ( S - C) ] ..... [ A,B, C are the triangle sides ]
Area = sqrt [ (15 + 6sqrt (3) ) * ( 15 + 6sqrt (3) - 13) * ( 15 + 6sqrt(3) - 17) * (15 + 6sqrt (3) - 12sqrt(3)) ] =
sqrt [ (15 + 6sqrt (3)) * (2 + 6sqrt(3)) * (6sqrt (3) - 2) * (15 - 6sqrt (3)) ] =
sqrt [ ( 6sqrt(3) + 2) * ( 6sqrt (3) - 2) * ( 15 + 6sqrt (3)) * ( 15 -6sqrt (3) ) ] =
sqrt [ (36*3 - 4) * (225- 36*3) ] =
sqrt [ 108 - 4) * ( 225 - 108) ] =
sqrt [ 104 * 117] =
sqrt [ 26 * 4 * 13 * 9] =
sqrt [ 13 * 2 * 4 * 13 * 9 ] =
sqrt [ 13^2 * 2^2 * 3*2 * 2 ] =
13 * 2 * 3 * sqrt (2) =
78 sqrt (2) units^2