Jim, you said


sqrt (R^2 + d^2) = R + h



I don't think so. If R = 1 and D = 1, then the root of the sum of the
squares isn't 2, but 1.414. .


but for sure distance to the horizon is ((R + d) ^2 - R ^2) ^.5

maybe you were thinking (R + d)^2 as R^2 + 2Rd + d^2, so that the root
is really of (2Rd +d^2)

This would be fun, remembering some of the approximations when Rd
(except in Mx's universe)