The past few decades have seen the development of a number of techniques for calculating scattering amplitudes without recourse to quantum fields and Feyn man diagrams. These are broadly referred to as on-shell methods. The devel opment of these methods has birthed a new perspective on scattering ampli tudes. In many cases they can be thought of as differential forms with special properties on constrained spaces - i.e. as differential forms on so called posi- tive geomet ries. The constraints of positivity suffice to completely determine the singularity structure of the S-matrix in these cases. This thesis is focused on understanding the emergence of known physics from positive geomet ries, computations in this new forma lism and developments in on shell methods for theories involving massive particles. The first two chapters study the amplituhe dron which is the positive geometry relevant to N = 4 super Yang-Mills. They focus on understanding the constraints of positivity and on seeing the physics of unitarity emerge from it. The next chapter develops geometric and on-shell techn ique s for understanding the properties of one loop integrals from the in tegrand in Feynman parameter space. The final chapter takes a step closer to the real world and develops an on-shell formalism for computing amplitudes in the bosonic, electroweak sector of the Standard Model. This includes an on shell understanding of the Higgs mechanism within the context of the Standard Model.