Is there any advantage of including all the non-linearities of a model in the objective function compared to writing the non-linearities as individual constraints?

Let me clarify that with a fictitious example:

Let us consider we want to minimise the area of a circle, so our decision variable is the radius of the circle. We can model this problem by writting the area directly in the objective function:

OF =e= pi * r²;

An alternative way is to calculate the area of the circle as an individual equality constraint and include the variable that stores the value of the area of the circle in the objetive function:

variable_area =e= pi * r²;

OF =e= variable_area;

The reason I am asking this question is that in The GAMS Model Library there is a model (synheat 117) in which all the non-linearities are included in the objective function. I modelled the same problem but wrote the non-linearities as individual equality constraints and included the variable of the non-linear constraints in the objetive function.

Thank you very much for your help on this.