Slender–body theory is used to determine the approximate static shape of a conically ended dielectric drop in an electric field. The shape and the electric–field distribution follow from solution of a second–order, nonlinear ordinary differential equation that can be integrated numerically or analytically. An analytic formula is given for the dependence of the equilibrium cone angle on the ratio of the dielectric constants of the drop and the surrounding fluid. A rescaling of the equations shows that the dimensionless shape depends only on a single combination of and the ratio of electric stresses and interfacial tension. In combination with numerical solution of the equations, the rescaling also establishes that, to within logarithmic factors, there is a critical field Emin for cone formation proportional to (−1)−5/12, at which the aspect ratio of the drop is proportional to (−1)1/2. Drop shapes are computed for E∞ > Emin. For E∞ ≫ Emin the aspect ratio of the drop is proportional to . Analogous results apply to a ferrofluid in a magnetic field.