Approximating surface areas by interpolations on triangulations

We consider surface area approximations by Lagrange and Crouzeix–Raviart interpolations on triangulations. For Lagrange interpolation, we give an alternative proof for Young’s classical result that claims the areas of inscribed polygonal surfaces converge to the area of the original surface under the maximum angle condition on the triangulation. For Crouzeix–Raviart interpolation we show that the approximated surface areas converge to the area of the original surface without any geometric conditions on the triangulation.