Earth is roughly 8000 miles in diameter.
(a) I'm riding in a hot air balloon 1 mile above the surface of earth. Approximately how far away is the horizon? (In other words, how far away is the farthest point on the surface of Earth that I can see.)
(b) What if I'm in a plane 6 miles above the surface of the earth?
(c) What is I'm in a spaceship 100 miles above the surface of the earth?
The radius of the Earth = 8000/2 = 4000 miles
We can figure the distance to the horizon using a right triangle
The distance from the center of the Earth to any object above the Earth's surface = the hypotenuse = 4000 + h where h is the height of the object [in miles ] above the Earth's surface
And the radius of the Earth is equal to one leg of the right triangle
And the distance from the object to the horizon, D, is the other leg
So...we have this general formula to determine D
D = √ [ (4000 + h)^2 - 4000^2]
(a) D = √ [ (4000 + 1)^2 - 4000^2 ] ≈ 89.4 miles
(b) D = √ [ (4000 + 6)^2 - 4000^2 ] ≈ 219.2 miles
(c) D = √ [ (4000 + 100)^2 - 4000^2 ] = 900 miles