Radius of Earth

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Hi purplec16,

The horizon makes a 90 degree angle with the Earth's centre.
This is because the astronaut's line of sight makes a tangent with the planet
(if you balance a perfectly round ball on a table, it's centre is directly above the point of contact or tangency).

The triangle has the same sides and angles,
so the trigonometry works exactly as skeeter originally outlined.

$Sin(65.8^o)=\frac{opposite}{hypotenuse}=\frac{R}{R +380}$

$(R+380)Sin(65.8^o)=R$

$RSin(65.8^o)+380Sin(65.8^o)=R$

$380Sin(65.8^o)=R-RSin(65.8^o)=R[1-Sin(65.8^o)]$

$R=\frac{380Sin(65.8^o)}{1-Sin(65.8^o)}$

Sorry Skeeter, my mistake 380 miles is correct

Thank you so much, I get it now and I completely understand, Thank you Skeeter and Archie