A common issue for watersports refers to the distance to the horizon. Often the question is asked, how far you can see a moving ship just beyond the horizon. We do suspect that this distance will depend on the vertical size of the ship and the observer's elevation above the sea surface. With the help of some simple mathematical considerations, applying not more than the Pythagorean theorem, we are able to answer this question. First of all, we should sketch the situation:

It shows the circular cross-section of the earth with its radius R and center M. We attach to this circle a tangent through the point H. Left on this line there is point P₁ located at a distance l₁ from H and with a height h₁ above the sea level. This point P₁ represents the ship with a height h₁. To the right side of H there is point P₂ localted at a distance l₂ of H and with a height h₂. This point P₂ represents the position of the observer. The triangle M, P₁, H can be described by the Pythagorean theorem, applying the following relationship:

By multiplying (binomial formula), we can write down the right side of the equation as:

Substracting the term R² on both sides, we get:

After applying the square root on both side, we obtain the equation for determining the distance from the horizon l₁:

For the entire distance l from P₁ tor P₂ we obtain finally:

The distance l thus corresponds to the maximum viewing distance of a vessel with a height h₁ and an observer with an elevation h₂ above the sea's surface. For reasons of symmetry, we can swap the roles of the ship and its observer. The following input form is calculating these three distances l₁,l₂ and l. Of course, this calculation needs to know the value for the earth's radius. This value is approximately 6370 km.