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_,m_]:=ParametricPlot3D[{Cos[θ] (1-z) m Radius[h],Sin[θ] (1-z) m 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,m],axes],{{h,0,"height"},0,1},{{m,1,"scale factor"},0.1,2}]

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_,m_,c_]:=ParametricPlot3D[{Cos[θ] (1-z) m Radius[h],Sin[θ] (1-z) m Radius[h],h z},{θ,0,Pi+h Pi},{z,0,1},PlotStyle->c,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,m,c],axes],{{h,0,"height"},0,1},{{m,1,"scale factor"},0.1,2},{{c,Blue,"color"},Blue,Red,Green}]

Radius[h_]:=r/.
First@Solve[{Pi r Sqrt[r^2+h^2] (1+h)/2==Pi/2,h>0},r,Reals];
Radius[0]=1;

ta[h_]:=2 Pi-2 Pi Sqrt[r^2-(r Sin[beta[h]])^2]

beta[h_]:=ArcSin[Sqrt[(4 Pi-ta[h]) ta[h]]/(2 Pi)]

AnimatedCone[h_]:=ParametricPlot3D[{Cos[θ] (1-z) Radius[h],Sin[θ] (1-z) Radius[h],h z},{θ,0,ta[h]+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}]

((Sqrt[3.5481432270250993`*^18 (η^2)-1.1294090667581471`*^18 (r^2) θ+8.987551787368176`*^16 (r^2) (θ^2)])/(Sqrt[39.47841760435743` (η^2)-12.566370614359172` (r^2) θ+(r^2) (θ^2)]))

Radius[h_]=r/. First@Solve[{Pi r+h^3 Sqrt[r^2+h^2] (1+h)/2==Pi/2,h>0},r,Reals];
Radius[0]=1;
AnimatedCone[h_,m_,c_]:=ParametricPlot3D[{Cos[θ] (1-z) ((√(-1.1294090667581471`*^18 θ+8.987551787368176`*^16 (θ^2)+3.5481432270250993`*^18 ((Sin[m])^2)))/((Sqrt[-12.566370614359172` θ+(θ^2)+39.47841760435743` ((Sin[m])^2)]))) Radius[h],Sin[θ] (1-z) m Radius[h],h z},{θ,0,Pi+h Pi},{z,0,1},PlotStyle->c,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,m,c],axes],{{h,0,"height"},0,1},{{m,1,"scale factor"},0.1,2},{{c,Blue,"color"},Blue,Red,Green}]

Radius[h_,r_,θ_]:=x/. First@Solve[{2 π r Sqrt[r^2-h^2] (1+h)/2==θ r,h>0,r>0,θ>0},x,Reals];
ConeAngle[h_,r_,θ_]:=β/. First@Solve[{β==ArcSin[Sqrt[(4 π-θ) θ]/(2 π)],h>0,r>0,θ>0},β,Reals];
AnimatedCone[h_,r_,θ_]:=ParametricPlot3D[{Cos[ϕ] (1-z) r Radius[h,r,θ],Sin[ϕ] (1-z) r Radius[h,r,θ],h z},{ϕ,0,Pi+(ConeAngle[h,r,θ])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,r,θ],axes],{{h,0,"height"},0,1},{{r,1,"initial radius"},0.1,2},{{θ,0,"θ (sector of the circle removed to form the base of the cone)"},0,2 π}]

Radius[h_,r_,β_]:=x/. First@Solve[{2 π r-2 π Sqrt[r^2-h^2]==θ r,ArcSin[Sqrt[(4 π h-θ h)/(2 π)]]==β,h>0,r>0,β>0,θ>0},x,Reals];
Radius[0,0,0]=1;
AnimatedCone[h_,r_,θ_,β_,m_]:=ParametricPlot3D[{Cos[θ] (1-z) m Radius[h,r,β],Sin[θ] (1-z) m Radius[h,r,β],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,r,θ,β,m],axes],{{h,0,"height"},0,1},{{r,1,"initial radius"},0.1,5},{{θ,0,"theta"},0.1,8 π},{{β,0,"beta"},0.1,π/2},{{m,1,"scale factor"},0.1,2}]

Radius[h_,β_]=r/. First@Solve[{2 π r-2 π Sqrt[r^2-h^2]==θ r,h>0,β==ArcSin[Sqrt[(4 π-θ) θ]/(2 π)]},r,Reals];
Radius[0]=1;
AnimatedCone[h_,m_,θ_,β_]:=ParametricPlot3D[{Cos[ϕ] (1-z) m Radius[h,β],Sin[ϕ] (1-z) m Radius[h,β],h z},{ϕ,0,β+h β},{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,m,θ,β],axes],{{h,0,"height"},0,1},{{m,1,"scale factor"},0.1,m2},{{θ,0,"θ value"},0,2 π},{{β,0.2,"β value"},0.1,π}]

Radius[h_]=r/. First@Solve[{4 π r^2 θ-r^2 θ^2==2 π^2 h^2,h>0},r,Reals];
Radius[0]=1;
AnimatedCone[h_,m_,θ_,β_]:=ParametricPlot3D[{Cos[θ] (1-z) m Radius[h],Sin[θ] (1-z) m Radius[h],h z},{θ,0,2 π β},{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,m,θ,β],axes],{{h,0,"height"},0,1},{{m,1,"scale factor"},0.1,2},{{θ,0,"sector angle"},0,Pi},{{β,0,"angle between hypoteneuse and base"},0,1}]