ParametricPlot3D[{x, y, x*y}, {x, -1, 1}, {y, -1, 1}] was a popular shape for "modernistic" public attractions in former decades--a glass walled, saddle shaped shell standing on two corners. ("Times table architecture".) What's its area? (Answer at end.) Now ParametricPlot3D[{{x, y, x*y}, {x, Sqrt[2]*Cos[Pi/4*x]*y, Sqrt[2]*Sin[Pi/4*x]*y}}, {x, -1, 1}, {y, -1, 1}] . Do these surfaces really coincide? What should the Cos and Sin really be? Now try ParametricPlot3D[{{x, y, x*y},r*x*{Cos[Pi*y/2], Sin[Pi*y/2], r*x*Sin[Pi*y]/2}}, {x, -1, 1}, {y, -1, 1}] for various 1<r<2. What are the areas of these round ones? --rwg Ans 1: A surprisingly complicated 4/Sqrt[3] - 2*Pi/9 + 6*ArcCsch[Sqrt[2]] + 2/3*Log[(Sqrt[3]-1)/Sqrt[2]] ~ 5.12316 . Ans 2: 1/Sqrt[1+x^2], x/Sqrt[1+x^2], Ans 3: 2 2 3/2 - Pi ((1 + r ) - 1) . Scales as r^3? 3 Also check out ParametricPlot3D[{{x, Sqrt[2]/Sqrt[1 + x^2]*y,Sqrt[2]*x/Sqrt[1 + x^2]*y}, {-Sqrt[2]/Sqrt[1 + x^2]*y, x, -Sqrt[2]*x/Sqrt[1 + x^2]*y}}, {x, -3, 3}, {y, -3, 3}] Voices tell me that the Shanghai-based Clank foundation is offering big bucks for the capture and conviction of the fabled parabolic hyperboloid--hyperbolas in sections parallel to two coordinate planes and parabolic intersections with the 3rd set of planes. Omigod, I've been using Vaseline five years past expiration!