$Assumptions={0<c<1}

S[c_]=2Pi Integrate[f[z]Sqrt[1+f'[z]^2]/.f->Function[q,rc[q,c]],{z,-hc[c],hc[c]}]

Plot[S[c],{c,0,1}]

NMinimize[{InverseFunction[hc][h],1>=c≥.2},c]

hc[c_]:=c ArcCosh[1/c]
rc[r_,c_]:=c Cosh[r/c]

Plot[hc[c],{c,0,1}]

D[hc[c],c]//FullSimplify;
ccrit=FindRoot[%,{c,.5}][[1,2]];
hcrit=hc[ccrit];
cmin=.01;
cmax=.99;

Manipulate[
Module[{ch},
ch={
FindRoot[hc[c]-h,{c,cmin,cmin,ccrit},MaxIterations->1000],
FindRoot[hc[c]-h,{c,cmax,ccrit,cmax}]
}[[All,1,2]];
GraphicsArray[{{ControlActive[
Show[
ParametricPlot3D[{r,0,rc[r,#]}&/@ch//Evaluate,{r,-h,h},PlotStyle->Directive[Thick,Blue]],
ParametricPlot3D[{#,Sin[th],Cos[th]}&/@{-h,h},{th,0,2Pi},PlotStyle->Directive[Thick,Blue]],
PlotRange->{{-.7,.7},{-1,1},{-1,1}},
Axes->{True,False,False},
AxesOrigin->{0,0,1},
AxesStyle->Thick,
Boxed->False
],
Show[RevolutionPlot3D[
Evaluate[rc[r,#[[1]]]],{r,-h,h},{th,.4Pi Boole[cut],2Pi},
PlotRange->{{-.7,.7},{-1,1},{-1,1}},
RevolutionAxis->{1,0,0},
Mesh->None,
PlotStyle->#[[2]],
BoundaryStyle->Directive[Thick,Blue],
RotationAction->"Clip",
PlotPoints->50,
Boxed->False,Axes->False]&/@
{{ch[[1]],Directive[Red,Opacity[1],Specularity[White,1000]],.4Pi},
{ch[[2]],Directive[Green,Opacity[1],Specularity[White,1000]],.4 Pi}}]],
ParametricPlot[{{hc[c],S[c]},{h,3Pi c},{3 Pi c,2Pi}},{c,0,1},
PlotRange->{{0,1},{0,3Pi}},
AspectRatio->1,
PlotLabel->"Pole powierzchni",
Epilog->{PointSize[Large],Green,Point[{h,S[ch[[2]]]}],Red,Point[{h,S[ch[[1]]]}]}]
}}]],
{h,Max[hc[cmin],hc[cmax]]//Evaluate,hcrit},
{{cut,False,"Przekrój"},{True,False}}]