Many microwave (RF) circuit designs require passive distributed sub-components with prescribed scattering parameters. These sub-components have typically been realised by cascading building-block configurations (eg. transmission lines of specific lengths, bends in transmission lines, coupled lines, and so on) of standard shape, and then adjusting the dimensions of selected prescribed features of these building-blocks. The problem with this approach is that the resulting sub-component may take up more "real-estate" on the overall circuit board than can be tolerated, may require tolerances that are too tight and hence be more costly than product developers can allow, can lead to less-than-best performance because we select the building-blocks (that we think are needed) ahead of time, and so on. The research in this thesis contributes to the shape synthesis approach of physical microstrip circuit design. The shape synthesis process is usually contrasted to traditional design by recognizing that it does not merely adjust the dimensions of a set of prescribed geometrical features on pre-selected shapes, but allows the electromagnetic physics to tell us what the sub-component layout needs to be (and it can be unconventional) in order to obtain the required performance. Existing shape synthesis techniques are based on the discrete- or continuous-pixelation method. Each of these approaches, however, have disadvantages (eg. too many degrees of freedom required to achieve the geometrical resolution necessary; the need for arbitrary decisions to fix material properties) that have prevented shape synthesis from being accepted for widespread use in design practice. In this thesis we develop, implement and apply a completely new shape synthesis approach, called the subtractive approach, that overcomes many of the above-mentioned disadvantages of pixelation-based methods It reduces the number of variables (degrees of freedom) needed in spite of the fact that the "design space" is significantly broadened by this approach. The latter is confirmed by the fact that it produces physical circuit geometries that we would not have come up with using traditional design methods. Examples are provided of the application of the new subtractive shape synthesis method. This new method involves continuous variables directly related to the physical circuit geometry, and thus could be used with surrogate modelling, unlike some existing shape synthesis procedures.