Hi, I am probably going to sound like one of those old people who type Google searches into facebook .... I now understand your question, you are asking about how to induce a metric from an existing embedding of a smooth real surface into Euclidean space. If you start with a surface which is not embedded into Euclidean space the Riemannian metrics form an uncountable set that is easy to calculate; you are dealing with how to choose one particular Riemann metric. And there is no issue with how any Riemannian metric has an underlying two-form whose integral gives the area. Also if you interpret a Riemannian metric as an element of the second symmetric power of differential one-forms, the conormal bundle has a natural embedding into the one-forms of the Euclidean space restricted to the surface, and the quotient is isomorphic to the one-forms of the surface. You apply the induced map on second symmetric powers to the element which defines the Riemann metric of the Euclidean space, and this gives an element of the second symmetric power of the one-forms of the surface, which is your Riemannian metric of the surface.