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Clear["Global`*"]; k = 4; ptss = Table[ TranslationTransform[{0, 0, \[Theta]}]@* ScalingTransform[ Sqrt[3^2 - \[Theta]^2]/(2 Sin[\[Pi]/k])*{1, 1, 1}]@* RotationTransform[0, {0, 0, 1}] /@ (PadRight[#, 3] & /@ CirclePoints[k]), {\[Theta], Subdivide[3, 0, 600]}]; {m, n, p} = Dimensions[ptss]; indexes = Most@Partition[ ArrayReshape[Range[m*n], {m, n}], {2, 2}, {1, 1}, {1, 1}]; bm = BoundaryMeshRegion[ Flatten[ptss, 1], {Rest@ Map[Polygon[{#[[1, 1]], #[[1, 2]], #[[2, 2]]}] &, indexes, {2}], Map[Polygon[{#[[2, 1]], #[[2, 2]], #[[1, 1]]}] &, indexes, {2}], Polygon[Range[m*n - n + 1, m*n]]}] 

Clear["Global`*"]; k = 6; pt[θ_] := PadRight[#, 3, θ] & /@  CirclePoints[{Sqrt[3^2 - θ^2]/(2 Sin[π/k]), 0}, k]; lines = ParametricPlot3D[pt[x], {x, 0, 3}, BoxRatios -> Automatic]; surface = ParametricPlot3D[{1 - s, s} . # & /@ Partition[pt[x], 2, 1, 1] // Evaluate, {x, 0, 3}, {s, 0, 1}, Boxed -> False,  PlotStyle -> Opacity[.2], Axes -> False, PlotPoints -> 60,  MaxRecursion -> 0, Mesh -> None]; Show[lines, surface, Graphics3D[{FaceForm[Directive@{Opacity[.5], Yellow}], Polygon[pt[1.5]], FaceForm[], EdgeForm[Directive@{Orange, Thick}],  Polygon[pt[0]]}], Boxed -> False, Axes -> False] 

Clear["Global`*"]; k = 6; L[θ_] := Sqrt[5^2 - θ^2]/(2 Sin[π/k]); range = FunctionDomain[L[θ], θ]; pts[θ_] := PadRight[#, 3, θ] & /@ CirclePoints[{L[θ], 0}, k]; lines = ParametricPlot3D[ pts[x], {x, Mean[{range[[1]], range[[-1]]}], range[[-1]]},  BoxRatios -> Automatic]; surface = ParametricPlot3D[{1 - s, s} . # & /@ Partition[pts[x], 2, 1, 1] // Evaluate, {x, Mean[{range[[1]], range[[-1]]}], range[[-1]]}, {s,  0, 1}, Boxed -> False, PlotStyle -> Opacity[.2], Axes -> False,  PlotPoints -> 60, MaxRecursion -> 0, Mesh -> None]; Show[lines, surface, Graphics3D[{FaceForm[Directive@{Opacity[.5], Yellow}], Polygon[pts[1.5]], FaceForm[], EdgeForm[Directive@{Orange, Thick}],  Polygon[pts[0]]}], Boxed -> False, Axes -> False, PlotRange -> All] 

Clear["Global`*"]; k = 6; pt[θ_] := TranslationTransform[{0, 0, θ}]@*  ScalingTransform[   Sqrt[3^2 - θ^2]/(2 Sin[π/k])*{1, 1, 1}]@*  RotationTransform[0, {0, 0, 1}] /@ (PadRight[#, 3] & /@ CirclePoints[k]); lines = ParametricPlot3D[pt[x], {x, 0, 3}, BoxRatios -> Automatic]; surface = ParametricPlot3D[{1 - s, s} . # & /@ Partition[pt[x], 2, 1, 1] // Evaluate, {x, 0, 3}, {s, 0, 1}, Boxed -> False, PlotStyle -> Opacity[.2], Axes -> False, PlotPoints -> 60, MaxRecursion -> 0, Mesh -> None]; Show[lines, surface, Graphics3D[{FaceForm[Directive@{Opacity[.5], Yellow}], Polygon[pt[1.5]], FaceForm[], EdgeForm[Directive@{Orange, Thick}], Polygon[pt[0]]}], Boxed -> False, Axes -> False] 

Clear["Global`*"]; k = 6; pt[θ_] := PadRight[#, 3, θ] & /@ CirclePoints[{Sqrt[3^2 - θ^2]/(2  Sin[π/k]), 0}, k]; lines = ParametricPlot3D[pt[x], {x, 0, 3}, BoxRatios -> Automatic]; surface = ParametricPlot3D[{1 - s, s} . # & /@ Partition[pt[x], 2, 1, 1] // Evaluate, {x, 0, 3}, {s, 0, 1}, Boxed -> False, PlotStyle -> Opacity[.2], Axes -> False, PlotPoints -> 60, MaxRecursion -> 0, Mesh -> None]; Show[lines, surface, Graphics3D[{FaceForm[Directive@{Opacity[.5], Yellow}], Polygon[pt[1.5]], FaceForm[], EdgeForm[Directive@{Orange, Thick}], Polygon[pt[0]]}], Boxed -> False, Axes -> False]