Recently, I have come across a map showing the coverage of a missile given its radius on a Mercator projection. In a 3-D space, the boundary of the missile would be a circle of a sphere. However, I am curious as to how this is translated onto the 2-D Mercator projection. Note that the launch is from around North Korea, symbolised by the triangle, with a radius of 8000km.
Assuming that the Earth is perfectly spherical, and that it has a radius of $1$, if I have the latitude ($φ$) and longitude ($θ$) of a location, and the range $d$ of the missile, how would I find the formula giving me the 2-D graph upon the Mercator projection (in the form $y=f(x)$)? Currently, I am in the last year of secondary education (I would highly appreciate it if the method did not extend beyond this level too much).