ClearAll[AnimatedCone, Radius]
Radius[h_] = r /. First@
    Solve[{Pi r Sqrt[r^2 + h^2] (1 + h )/2 == Pi /2, h > 0}, r, Reals];
Radius[0] = 1;
AnimatedCone[h_] := ParametricPlot3D[{Cos[θ] (1 - z) Radius[h], Sin[θ] (1 - z) Radius[h], h z},
   {θ, 0, Pi + h Pi}, {z, 0, 1}, 
   PlotStyle -> Blue, 
   MeshFunctions -> {#2&},
   Boxed -> False, 
   PlotRange -> 1.5 {{-2, 2}, {-2, 2}, {0, 1}}, 
   BoundaryStyle -> GrayLevel[.1], 
   PerformanceGoal -> "Quality", 
   Axes -> False];
axes = Graphics3D[{Green, Arrowheads[Medium], 
    MapThread[{Arrow[{{0, 0, 0}, 1.4 #2}], Text[#, 1.5 #2]} &, 
     {{"X", "Y", "Z"}, IdentityMatrix[3]}]}];
Manipulate[Show[AnimatedCone @ h, axes], {{h, 0, "height"}, 0, 1}]

ClearAll[AnimatedCone,Radius,θ,x,β]
Radius[x_,h_]=r/. First@Solve[{2 π r-2 π Sqrt[r^2-h^2]==x-h,h>0},r,Reals];
Radius[x_,0]=x;
θ[x_,h_]=(2 (π r-π x))/r/. First@Solve[{2 π r-2 π Sqrt[r^2-h^2]==x-h,h>0},r,Reals];
β[h_]=ArcSin[Sqrt[(4 π-θ[x,h]) θ[x,h]]/(2 π)];
AnimatedCone[x_,h_]:=ParametricPlot3D[{Cos[θ[x,h]] (1-z) Radius[x,h],Sin[θ[x,h]] (1-z) Radius[x,h],h z},{θ[x,h],0,π+h π},{z,0,1},PlotStyle->Blue,MeshFunctions->{#2&},Boxed->False,PlotRange->1.5 {{-2,2},{-2,2},{0,1}},BoundaryStyle->GrayLevel[.1],PerformanceGoal->"Quality",Axes->False];
axes=Graphics3D[{Green,Arrowheads[Medium],MapThread[{Arrow[{{0,0,0},1.4 #2}],Text[#,1.5 #2]}&,{{"X","Y","Z"},IdentityMatrix[3]}]}];
Manipulate[Show[AnimatedCone@x,axes],{{x,2,"Base"},0.2,3},{{h,0,"height"},0,3},{{α,0,"angle"},0,2 π},ControlType->Slider]