  A family of embedded rotationally symmetric tori in the Euclidean 3-space
consisting of two opposite signed constant mean curvature surfaces that
converge as varifolds to a double round sphere is constructed. Using complete
elliptic integrals, it is shown that their Willmore energy lies strictly below
$8\pi$. Combining such a strict inequality with previous works by
Keller-Mondino-Rivi\`ere and Mondino-Scharrer allows to conclude that for every
isoperimetric ratio there exists a smoothly embedded torus minimising the
Willmore functional under isoperimetric constraint, thus completing the
solution of the isoperimetric-constrained Willmore problem for tori. Similarly,
we deduce the existence of smoothly embedded tori minimising the Helfrich
functional with small spontaneous curvature. Moreover, it is shown that the
tori degenerate in the moduli space which gives an application also to the
conformally-constrained Willmore problem. Finally, because of their symmetry,
the Delaunay tori can be used to construct spheres of high isoperimetric ratio,
leading to an alternative proof of the known result for the genus zero case.
