Isometry between triangle and cone

Question

In a certain exercise I have been asked to say if there is an isometry between the triangle $T =\{ z = 0, 0 < x, y, x + y < 1 \}$ and the cone $C = \{x^2 + y^2 =\frac{z^2}{4}, 0 < z < 2\}$. I tried with a parametrization of the cone $(x,y,z) = (r\sin \theta, r \cos \theta, 2r)$. The parametrization of the plane seems much diffuclt using these coordinates, and I don't know how to continue. I would really appreciate if somebody could give me a hint of how to deal with this problem. I think that they are isometric, but I might be wrong.