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On a clear day: solution
R is the radius of the Earth, H is your height and D is how far you can see. The walking distance across the Earth's surface is the length of the arc s.
Solution to problem 1: In Issue 54's Outer Space you were asked to work out the "walking" distance over the Earth's surface to the horizon. This is equal to the length of the arc of the circle from the point where you are to the point where your line of sight hits the surface of the Earth, which is
Solution to problem 2: You see something, like a cloud or a hot air balloon, high in the sky beyond the horizon and you know its maximum possible height above the ground. Can you work out how far away it could be?
Let's assume that the object is at the maximum possible height above the surface of the Earth. So in our two-dimensional view it lies somewhere on the circle of radius about the centre of the Earth. Since it is beyond the horizon, the object lies beyond (from your point of view) the straight line through and the horizon point . And since it is visible over the horizon, the object lies "above" the tangent line passing through your view point and the horizon point . Thus the object lies somewhere on the arc of the circle between points and (see the diagram on the right).
Using Pythagoras' theorem, we see that the distance from you to point is
Now let be the distance from to . Using Pythagoras' theorem on the right-angled triangle , we get