The influence of free-stream conicity on flow over a slender sharp cone was investigated. A unified conicity parameter d is defined, both theoretical analysis and numerical simulation were conducted for a range of flow conditions. Results show that conicity can strongly alter surface pressure, boundary-layer thickness, edge density, and wall heat flux compared to parallel flow, while the shock shape is less sensitive. Inviscid analysis reveals that the conicity response is governed by the hypersonic similarity parameter K and streamwise coordinate S, normalized distributions for identical K collapse into same curve, indicating that conicity acts primarily as a streamwise scaling. At high Mach numbers, strong viscous interaction modifies the effective body shape and degrades the accuracy of analytical predictions, disrupting the scaling collapse. Thermochemical nonequilibrium has a minor effect on conicity corrections, although species distributions do not follow the scaling because they are controlled by finite rate chemistry. Empirical transition assessment shows that conicity can delay or even suppress transition by increasing the transition Reynolds number and reducing the local edge Reynolds number. Since small conicity significantly affects the flow over slender cones, it is essential to first quantify the conicity level in experiment. If the conicity is non-negligible, its effects must be accounted for; when strong viscous interaction invalidates analytical scaling, CFD corrections should be introduced. The conicity metric is not restricted to conical nozzles and can be extended to other divergent free-streams using the Mach number gradient.