Find the area of a triangle with side lengths

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