Estimating Distance While Paddling
Practical Geometry for Kayakers & Canoeists
by Wes Kisting
However flat the earth may look, we all know that it's actually round. Because of this fact, distant objects vanish over the horizon. When we paddle toward them, we see the tops of those objects emerge into view before we see their bottoms. The same is true as those objects move away from us: their bottoms vanish over the horizon before their tops. Kayakers who are attentive to this phenomenon (and willing to do some simple math) can use this knowledge to gauge distance more accurately.
Curvature of the Earth: Distance
The three pictures below show the position of an island in relation to the horizon. In the first picture, the bottom edge of the island is over the horizon. In the second picture, the island is much closer, but its bottom still has not crossed over into view. In the third picture, the bottom of the island has finally crossed over so that its sandy shoreline is now just barely visible. At this point, the kayaker can safely conclude that the island is 1.5 miles away.
[Editor's Note: Images are not shown to scale. The island has been enlarged in all three pictures to help make its relative position to the horizon easier to see on a monitor.]
How can we tell that the shore is 1.5 miles away in the third picture? By combining some simple geometry with our knowledge of the earth's curvature. Let's call this Distance Rule 1:
The distance from the paddler to the horizon (in miles) is equal to the square root of the paddler's height at eye level (in feet).
For a typical paddler sitting upright in a kayak, eye level will be approximately 2 feet above the water. If we take the square root of 2, we find that the horizon should be located 1.4 miles away (call it 1.5 for the sake of convenience). The exact distance will be slightly less for shorter paddlers and slightly more for taller paddlers, but unless you are exceptionally short or tall, the difference will be negligible. Likewise, if you are sitting significantly higher, such as in a canoe, the distance to the horizon will be noticeably greater. Thus, for a canoeist who is sitting so that his eyes are 3 feet above the water, the distance to the horizon will be 1.7 miles (an extra quarter-mile). If you like, you can measure the height of your eye level while sitting upright in your canoe or kayak, then calculate its square root to come up with an exact distance, in miles, to the horizon's vanishing point.
Why is it useful to know the horizon distance formula? Well, for one thing, it is now possible to reliably estimate your distance from shores and objects by using the horizon. As the base of an object passes into view from over the horizon, it is 1.5 miles away. That translates to a half-hour of paddling at a typical cruising pace of 3 miles per hour. Anything over the horizon is more than a half-hour away; anything closer than the horizon is less than a half-hour away. This knowledge comes in handy when you're keeping an eye on weather patterns and trying to predict how soon you can be off the water.
Expanding the Distance Rule: Beyond the Horizon
We can expand on the same formula to estimate the distance from the horizon to an object (of known height) which is visible over the horizon, even before its base comes into view. Imagine paddling across the sea toward a distant island: We'll see the top of the island long before we see the base. In fact, if the island is very tall, we might see its top days before its base comes into view. If we know the approximate height of the island, however, we can calculate its distance beyond the horizon as soon as its top comes into view. Let's call this Distance Rule 2:
The distance from the horizon to an object just emerging over the horizon (in miles) is equal to the square root of the object's height (in feet).
Let's say that we are paddling toward an island that is known to be 100 feet tall at its highest point. Distance Rule 2 tells us that the tip of the island will just begin to emerge over the horizon when the island is 10 miles beyond the horizon (The square root of 100 is 10). To calculate our total distance from the island, however, we also need to factor in the distance from the horizon to us, which, as we know from Distance Rule 1, is about 1.5 miles. So, the tip of the island will just begin to peek into view when we are 11.5 miles away (10 + 1.5 = 11.5 miles).
Putting it all Together: To the Horizon and Beyond
In case you missed it, we just combined our first two distance rules to calculate the total distance from the paddler to an object emerging over the horizon. Let's call this Distance Rule 3:
The total distance from the paddler to an object just emerging over the horizon (in miles) is equal to the square root of the object's height (in feet) plus the square root of the paddler's height at eye level (in feet).
Because we know that the paddler's height at eye level is typically about two feet high and this value remains constant, we don't actually need to recalculate this part of the equation each time we apply the rule. So, the simplified version of Distance Rule 3 could read:
The total distance from the paddler to an object just emerging over the horizon (in miles) is equal to the square root of the object's height (in feet) plus 1.5 miles.
The diagram below will help clarify this principle. Notice that because of the curvature of the earth, the paddler is just beginning to see the tip of the island he is paddling toward. He cannot see its base. Only the very tip of the island is visible, and from his point of view, it is virtually touching the horizon. This is the moment when the paddler can most accurately calculate the total distance from himself to the island.
Distance Rule 1 gives us the paddler's "Distance to Horizon" (shown in green). Distance Rule 2 gives us the horizon's "Distance to Emerging Object" (shown in orange). Distance Rule 3 simply adds the results of these two rules together to calculate "Total Distance" (shown in blue).
Returning to our earlier example of the 100-foot tall island, we added the square root of the island's height (10) to the square root of the paddler's height at eye level (1.5) to discover that a 100-foot tall island must be approximately 11.5 miles away when its tip first begins to emerge over the horizon (as shown in the diagram). By extension, if more than the tip of the island is visible above the horizon, the island must be closer than 11.5 miles away; or if the tip is not visible at all, the island must be further than 11.5 miles away.
Obviously, to apply these rules you need to know an approximate height for the object you are paddling toward. Fortunately, many nautical charts and topographical maps provide this information. A prudent navigator will also research this information prior to the trip and make height annotations on the map for later reference.
Estimating Distance by Guessing Height
Even if the exact height of an object on the horizon is not known, we can still apply our distance rules by estimating the height of familiar objects. For example, suppose that the trees in a particular paddling area are, on average, 20 feet tall. If you're paddling toward a flat island covered with those trees, you can pretty safely assume that the tips of the trees will come into view when the island is about 6 miles away. To come up with that number, I added the square root of the average tree height (4.5) to the square root of the paddler's height at eye level (1.5), for a total of 6 miles. Now, if those trees turn out to be sitting on top of a hill which is an additional 10 feet above the water, our calculations will turn out to be wrong and the trees will come into view when the island is actually about 7 miles away. Even so, our rough estimate would still be useful and reasonably accurate in spite of the error.
© 2007, Wesley Kisting