What is a reference that explains surface area as the limit of a sequence of the areas of polyhedral approximations? (Ideally, it would also explain the Schwarz Lantern.)
2026-04-13 14:04:07.1776089047
Reference Request - Surface Area as the limit of a sequence of areas of polyhedral approximations
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L. V. Toralballa wrote several articles on this question:
[1] Alpert, Louis I.; Toralballa, L. V. An elementary definition of surface area in $E^{n+1}$ for smooth surfaces. Pacific J. Math. 40 (1972), 261--268.
[2] Toralballa, L. V. A geometric theory of surface area. I. Non-parametric surfaces. Monatsh. Math. 74 (1970), 462--476.
[3] Toralballa, L. V. A geometric theory of surface area. II. Triangulable parametric surfaces. Monatsh. Math. 76 (1972), 66--77.
[3] Toralballa, L. V. Corrigenda: "A geometric theory of surface area. II. Triangulable parametric surfaces''. Monatsh. Math. 77 (1973), 156--157.
[4] Toralballa, L. V. A geometric theory of surface area. III. Non-triangulable parametric surfaces. Monatsh. Math. 77 (1973), 363--365.
For the Schwarz Lantern, you can look at
[5] Frieda Zames, Surface Area and the Cylinder Area Paradox, The Two-Year College Mathematics Journal 8, No. 4 (Sep., 1977), pp. 207-211