It seems that what you are asking is essentially how one can approximate the area of a surface via triangulations. This is a classical topic stretching back more than a century. The famous example of Schwartz Lantern shows that arbitrary triangulations do not work. Young first described conditions which triangulations should satisfy in order to approximate area:

W.H. Young, On the Triangulation Method of Defining the Area of a Surface, 1921.

These approximations have been studied extensively since then. For a description of Young's condition and other references see the following paper posted recently on the arXiv:

Kobayashi and Tsuchiya, Approximating surface areas by interpolations on triangulations, 2017.

See also

L. V. Toralballa, A geometric theory of surface area, 1970

for more references. I think that these papers answer your question. Some of these references are concerned only with 2-D surfaces, but Young's original paper seems to consider arbitrary dimensions. Further, these methods require very little regularity.

It seems that what you are asking is essentially how one can approximate the area of a surface via triangulations. This is a classical topic stretching back more than a century. The famous example of Schwartz Lantern shows that arbitrary triangulations do not work. Young first described conditions which triangulations should satisfy in order to approximate area:

W.H. Young, On the Triangulation Method of Defining the Area of a Surface, 1921.

These approximations have been studied extensively since then. For a description of Young's condition and other references see the following paper posted recently on the arXiv:

Kobayashi and Tsuchiya, Approximating surface areas by interpolations on triangulations, 2017.

See also

L. V. Toralballa, A geometric theory of surface area, 1970

for more references. I think that these papers answer your question. Some of these references are concerned only with 2-D surfaces, but Young's original paper seems to consider arbitrary dimensions. Further, these methods require very little regularity. I am pretty sure that the answer to your question is yes for surfaces in $R^3$, and I think that things should work in all other dimensions as well.

It seems that what you are asking is essentially how one can approximate the area of a surface via triangulations. This is a classical topic stretching back more than a century. The famous example of Schwartz Lantern shows that arbitrary triangulations do not work. Young first described conditions which triangulations should satisfy in order to approximate area:

W.H. Young, On the Triangulation Method of Defining the Area of a Surface, 1921.

These approximations have been studied extensively since then. For a description of Young's condition and other references see the following recent paper posted on the arXiv:

Kobayashi and Tsuchiya, Approximating surface areas by interpolations on triangulations, 2017.

See also

L. V. Toralballa, A geometric theory of surface area, 1970.

for more references. I think that these papers answer your question. Some of these references are concerned with only 2-D surfaces, but Young's original paper seems to consider arbitrary dimensions. Further, these methods require very little regularity.

It seems that what you are asking is essentially how one can approximate the area of a surface via triangulations. This is a classical topic stretching back more than a century. The famous example of Schwartz Lantern shows that arbitrary triangulations do not work. Young first described conditions which triangulations should satisfy in order to approximate area:

W.H. Young, On the Triangulation Method of Defining the Area of a Surface, 1921.

These approximations have been studied extensively since then. For a description of Young's condition and other references see the following paper posted recently on the arXiv:

Kobayashi and Tsuchiya, Approximating surface areas by interpolations on triangulations, 2017.

See also

L. V. Toralballa, A geometric theory of surface area, 1970

for more references. I think that these papers answer your question. Some of these references are concerned only with 2-D surfaces, but Young's original paper seems to consider arbitrary dimensions. Further, these methods require very little regularity.