I find it funny that you have to change from degrees to radians to make the math work how you want it to.
If you want to use natural units, you need to use radians, not degrees.
e.g. if you want to use the length of an arc to obtain the angle, as used in that formula, the most straightforward way is to use radians.
i.e. if you have a length along the curve of S, and have a radius of R, then the angle, in radians, is S/R.
If you want to use degrees, then you need to use 360 °*S/(2*pi*R) = 360 °/2pi is the conversion from radians to degrees.
When you put that exact equation in degrees, it comes out to approx 2.29
And how did you do that?
Did you just leave in S/R?
If so, what is this meant to represent?
Do you understand the formula being used?
Here is a simple diagram:
S is the length along the arc/hypothetical level surface, from the point of the observer to the point of the observed object.
R is the radius of Earth.
h0 is the height of the observer above Earth.
hh is the height hidden by Earth.
And for added bits, S0 and Sh break S into 2 pieces, the piece from the observer to the horizon, and the piece from the horizon to the object being observed.
a, a0 and ah are angles for S, S0 and Sh respectively.
i.e.
a = a0+ah
S = S0 + Sh
So what is the math, which holds regardless of if you are using radians or degrees?
Well, we have a right angle triangle such that:
cos(a0)=R/(R+h0)
And another so:
cos(ah)=R/(R+hh)
But we know R, and h0, but not a0, so we need to take the inverse.
This gives us:
a0 = arccos(R/(R+h0))
We don't directly know hh or ah, but we do have a relationship to get ah.
a = a0+ah
Rearranging this we get:
ah = a - a0.
And if we sub in the previous result we get:
ah = a - arccos(R/(R+h0))
Now we can rerrange the equation from the right angle triangle:
cos(ah)=R/(R+hh)
R+hh = R/cos(ah)
hh = R/cos(ah) - R
Now subbing in the result we had for ah we get:
hh = R/cos(a - arccos(R/(R+h0))) - R
But the question is what is a?
Well this depends on if you are using degrees or radians.
a is the angle subtended for an arc length of S.
If you are using radians, this is simple:
a = S/R.
If you are using degrees you instead need:
a = 360 ° * S/(2 * pi * R)
You cannot use a = S/R while using degrees for cos and arccos.
And the obvious giveaway that your reasoning is wrong is:
S/R=0.068
a0 = arccos(R/(R+h0)) = 1.60 °
S/R - arccos(R/(R+h0)) = -1.53 nonsense units.
So the angle to calculate the amount hidden would then be negative.
So congrats on showing you have no idea what you are talking about.
If instead you use the correct degrees method, where you use
a = 360 ° * S/(2 * pi * R)
then you end up with
a - arccos(R/(R+h0)) = 2.32 °,
and then eventually hh = 5.2 km
Before you try spewing some bs about how its not supposed to be in degrees, the below picture (which showed up in an earlier post on this thread) shows that the Radius of the earth was calculated with degrees.
No, it doesn't. And that is not the only formula you can use.
As such, you would need to multiply (or divide, I honestly don't remember) with pi to convert that number to be able to be used in radians instead of degrees.
No, you don't.
The conversion is 180/pi or pi/180, depending on how you go.