Some closed surfaces, their surface areas and volumes

Before reading this notebook evaluate the entire notebook: the shortcut  Alt+v+o  or the menu item:  

     Evaluation ⊳ Evaluate Notebook

A fortune cookie

ParametricPlot3D[{Cos[t], Cos[t+s],s},{t,0,2 π},{s,0,π},PlotPoints->{61,91},Mesh->None,PlotRange->{{-1.5`,1.5`},{-1.5`,1.5`},{0,4}},Axes->True,BoxRatios->{1,1,1}]

A Torus

Clear[Ra,ra]; Ra=4;ra=1;ParametricPlot3D[{Cos[th] (Ra+ra Cos[ph]),Sin[th] (Ra+ra Cos[ph]),ra Sin[ph]},{th,0,2 π},{ph,0,2 π},PlotPoints->{161,51},Mesh->False,PlotRange->{{-Ra-ra-0.5`,Ra+ra+0.5`},{-Ra-ra-0.5`,Ra+ra+0.5`},{-ra-0.5`,ra+0.5`}},Axes->False,BoxRatios->Automatic,ImageSize->400]

Clear[Ra,ra];  Manipulate[ParametricPlot3D[{Cos[th] (Ra+ra Cos[ph]),Sin[th] (Ra+ra Cos[ph]),ra Sin[ph]},{th,0,2 π},{ph,0,2 π},PlotPoints->{161,51},Mesh->False,PlotRange->{{-5-3-0.5`,5+3+0.5`},{-5-3-0.5`,5+3+0.5`},{-3-0.5`,3+0.5`}},Axes->False,BoxRatios->Automatic,ImageSize->400],{{Ra,2},2,5},{{ra,1},.1,3}]

The following surface is some kind of combination of a torus and a fortune cookie:

Clear[Ra,ra]; Ra=3;ra=1;ParametricPlot3D[{Cos[th] (Ra+ra Cos[ph]),Sin[th] (Ra+ra Cos[ph]),ra Cos[ph +2th]},{th,0,2 π},{ph,0,2 π},PlotPoints->{161,51},Mesh->False,PlotRange->{{-Ra-ra-0.5`,Ra+ra+0.5`},{-Ra-ra-0.5`,Ra+ra+0.5`},{-ra-0.5`,ra+0.5`}},Axes->True,BoxRatios->Automatic,ImageSize->600]

Range[1,6]

Clear[Ra,ra]; Manipulate[ParametricPlot3D[{Cos[th] (Ra+ra Cos[ph]),Sin[th] (Ra+ra Cos[ph]),ra Cos[ph +mm th]},{th,0,2 π},{ph,0,2 π},PlotPoints->{161,51},Mesh->False,PlotRange->{{-6-0.5`,6+0.5`},{-6-0.5`,6+0.5`},{-3-0.5`,3+0.5`}},Axes->False,BoxRatios->Automatic,Boxed->False,ImageSize->600],{Ra,1,4},{ra,.2,3},{{mm,2},Range[1,6],ControlType->Setter}]

Several cardioid tori

A simple cardioid torus

ParametricPlot3D[(3-(1+Cos[t])Cos[t]){Cos[θ],Sin[θ],0} +(1+Cos[t])Sin[t]{0,0,1},{t,0,2 Pi},{θ,0,2 Pi},PlotPoints->{101,41},Mesh->False,PlotRange->{{-5,5},{-5,5},{-1.5,1.5}},Axes-> True,ImageSize->600]

A simple cardioid torus with one rotation

nn=1;ParametricPlot3D[3{Cos[θ],Sin[θ],0}+(-(1+Cos[t])Cos[t])(Cos[nn θ]{Cos[θ],Sin[θ],0}+Sin[nn θ]{0,0,1}) +(1+Cos[t])Sin[t](-Sin[nn θ]{Cos[θ],Sin[θ],0}+Cos[nn θ]{0,0,1}),{t,0,2 Pi},{θ,0,2 Pi},PlotStyle->{Opacity[.8]},PlotPoints->{50,50},Mesh->False,PlotRange->{{-5.5,5.5},{-5.5,5.5},{-2,2}},Boxed->True,Axes-> True,ImageSize->600]

A simple cardioid torus with two rotation

nn=2;ParametricPlot3D[3{Cos[θ],Sin[θ],0}+(-(1+Cos[t])Cos[t])(Cos[nn θ]{Cos[θ],Sin[θ],0}+Sin[nn θ]{0,0,1}) +(1+Cos[t])Sin[t](-Sin[nn θ]{Cos[θ],Sin[θ],0}+Cos[nn θ]{0,0,1}),{t,0,2 Pi},{θ,0,2 Pi},PlotStyle->{Opacity[.8]},PlotPoints->{50,50},Mesh->False,PlotRange->{{-5.5,5.5},{-5.5,5.5},{-2,2}},Boxed->True,Axes-> True,ImageSize->600]

A simple cardioid torus with three rotations

nn=3;ParametricPlot3D[3{Cos[θ],Sin[θ],0}+(-(1+Cos[t])Cos[t])(Cos[nn θ]{Cos[θ],Sin[θ],0}+Sin[nn θ]{0,0,1}) +(1+Cos[t])Sin[t](-Sin[nn θ]{Cos[θ],Sin[θ],0}+Cos[nn θ]{0,0,1}),{t,0,2 Pi},{θ,0,2 Pi},PlotStyle->{Opacity[.8]},PlotPoints->{50,50},Mesh->False,PlotRange->{{-5.5,5.5},{-5.5,5.5},{-2,2}},Boxed->True,Axes-> True,ImageSize->600]

A simple cardioid torus with four rotations

nn=4;ParametricPlot3D[3{Cos[θ],Sin[θ],0}+(-(1+Cos[t])Cos[t])(Cos[nn θ]{Cos[θ],Sin[θ],0}+Sin[nn θ]{0,0,1}) +(1+Cos[t])Sin[t](-Sin[nn θ]{Cos[θ],Sin[θ],0}+Cos[nn θ]{0,0,1}),{t,0,2 Pi},{θ,0,2 Pi},PlotStyle->{Opacity[.8]},PlotPoints->{50,50},Mesh->False,PlotRange->{{-5.5,5.5},{-5.5,5.5},{-2,2}},Boxed->True,Axes-> True,ImageSize->600]

Your tasks

For each of the surfaces above determine the exact, or if that is not possible the approximate, value for

The surface area

The volume