A flexible rectangle sheet size $(a,b),a>b $ is folded half along side $a$ and glued to make a circular cone cut segment of vertex angle $60^{\circ}$ as shown with three edges $(b,a,b).$
( $60^{\circ}$ choice for cone apex angle deformation arises due to maximum volume created by internal pressure at $90^{\circ}$ corner obtained by maintaining second order continuity along a line perpendicular to glue line.)
After bending distorted edges $(a,b)$ are curved/mapped as conical helices with Clairaut minimal radii nearer to cone vertex as $ (r_a,r_b)= (a/4,b).$ The cone surface is a single boat shaped nappe.
Calculate bent area to verify $A= ab $ conserved due to isometry.
Calculate volume enclosed by parallel displacement of edge $AB$ (skew perpendicular to cone axis) along the helices.
It refers to Jack D'Aurizio A4 paper sheet bent volume problem with two nappes.
You can see below a diagram of the solid, made with GeoGebra. Green surface $ABC$ is described by $AB$ sliding along the border: it is apparent from the diagram that segment $CM$ lies over the surface, hence the solid is not convex.
The equation for the lateral surface (orange) can be obtained by noticing that a generic point $P$ on the sheet of paper, having distance $r$ from the midpoint $V$ of the folded side and forming an angle $\angle CVP=\theta$, gets mapped to point $$ P'=\left({1\over2}r\cos2\theta,\ {1\over2}r\sin2\theta,\ {\sqrt3\over2}r \right) $$
of the cone, if we take its axis along $z$ and $V$ as origin.