(* We compute the surface area of triangle ABC *) 
a={0,0,0}
b={2,0,0}
c={0,3,5}

(* parametrization *)
r=v(u a+(1-u)b)+(1-v)c

(* normal vector *)
ru=D[r,u]
rv=D[r,v]
n=Simplify[Cross[ru,rv]]
nn=Simplify[Norm[n],Assumptions-> 0<u<1 &&0<v<1]

(* surface area *)
Integrate[Integrate[nn,{u,0,1}],{v,0,1}]

(* Alternate parametrization *)
r=u a+v b +(1-u-v) c
ru=D[r,u]
rv=D[r,v]
n=Simplify[Cross[ru,rv]]
nn=Simplify[Norm[n],Assumptions-> 0<u<1 &&0<v<1]
Integrate[Integrate[nn,{u,0,1-v}],{v,0,1}]