How to Calculate the Distance to the Horizon
Calculating the distance to the horizon takes a bit of clever math. Not necessarily complicated, but not intuitive either. Consider what you’re trying to figure out, after all. You’re looking out from your boat across the water to the horizon line. And though it looks like a flat run from you to the edge of the planet, you know that’s not true. The water you’re looking at is actually curving into the distance with the shape of the Earth. It’s a little mind bending when you think about it.
So, how do we determine the distance from where we’re standing to the horizon? If you have an app on your phone or even an old almanac it may not be so hard. But let’s say you don’t have it. Well, the first thing you need to know is where you’re standing. Not all of us stand the same, right? And you’ll be using the height of your eyes as a guide, since that’s the point you’re measuring from. The calculation will therefore be different for a person standing on the deck of a fishing trawler compared to someone sitting in a kayak.
The one piece of information you need going into this is the radius of the Earth itself. For our calculation we’re going to use 3,958.8 miles.
The full method of achieving this formula is needlessly complicated but know that, when using the proper radius of the Earth, you can get a simple formula for determining distance to horizon. That formula is:
That means 1.22459 (a number we derive from knowing the radius of the earth and using the Pythagorean Theorem) times the square root (√) of your eye height (h). This calculation uses a fairly precise measurement for the earth’s radius and will give you very accurate numbers. If you’re curious about how the entire equation is developed, you can check it out on this site here.
Many sites that offer up similar calculations tend to round their numbers up. This is fine, of course, the difference between 2.8 miles and 3 miles when you’re eyeballing the horizon isn’t all that significant. But for the purposes of accuracy we’re giving you some solid, if longer, numbers.
So let’s say you’re sitting in your boat and your eyes are 3 feet above the surface of the water. The formula becomes:
The square root of three is 1.73205080757. So now the formula becomes:
1.22459 x 1.73205080757
So the distance to horizon, based on this, is 2.12105209844 miles.
If you’re standing up and you’re about 6 feet tall, let’s say your eyes are sitting at about 5.5 feet. That means;
1.22459 X 2.34520787991
That equals 2.87191811766 or nearly 3 miles off.
Things to Remember
Calculating square roots in your head isn’t always the easiest thing to do. That’s why having your phone handy for the calculations is ideal. And, as you can see, the distance increases the greater the height. That means standing on the deck of a massive cruise liner will offer you a distance to horizon far greater than what you’ll get sitting in a canoe.
One thing you’ll want to remember here is converting to decimals. This is important for height. In our second example we used 5.5 feet. That works out to 5’6”. But if your eye line is higher, say 5’9” you’ll need to convert to decimal. So it’s not 5.9 feet, it’s 5.75 feet.
Our previous distance to horizon calculation involved a similar equation however the number used was 1.17 rather than 1.22459. Some folks asked where that number came from. We looked into it and, because this is such a prevalent issue, we’ll pass it on to you.
1.17 miles comes from the USCG Light List. This list, published annually by the Coast Guard, details all lighthouses, sound signals, beacons, buoys and other aids to navigation in a Coast Guard area. At the beginning of these lists you will find general information. In that general information you’ll find a calculation to determine the distance a person is able to see an object on the horizon. Their calculation is 1.17 X the square root of your eye height.
So there are two issues with this equation.
1 – What does 1.17 refer to? There is no indication from what source that number was derived. In our new calculation we show you how that number is determined by way of the radius of the Earth itself. The 1.17 is unknown.
2 – Is this actually a calculation to the horizon? The equation, as written, seems to indicate that it will determine how far you can see something, not necessarily where the horizon is. Those two may be and likely often are the same, but not always. However, without knowing how anyone came up with that 1.17 number, it’s a moot point.
So, if you see a calculation anywhere that offers up that number, take it with a grain of salt. The calculation will not offer a mathematically correct way to determine the distance to the horizon from your current point of view.
What About Maximum Visibility?
If you watch the weather report on TV, you may hear the local weather person mention maximum visibility. On a foggy day it may be very minimal, less than a quarter of a mile. On a clear day they may say upwards of 10 miles. But how can that be if the distance to horizon when you’re standing on shore may be less than 3 miles?
The thing you need to remember here is that maximum visibility is not about calculating where the horizon is. Instead, it refers to the ability to see and identify a prominent dark object against the sky at the horizon during the day. This is all related to the opacity of the atmosphere. So if it’s a hazy day, a smoggy day, or a rainy day, the visibility will decrease. The horizon is irrelevant in this case.
A Note of Thanks
Special thank you to Boatsafe visitor Robert Gillies who, among a couple of others, noticed our previous calculations were not entirely correct and pointed us in the direction of some superior math. Math is not always everyone’s strong suit but it’s nice to know how to use it when and where you need it! Once we were on the right track we discovered many sites were using that same calculation. And since it was derived from the United States Coast Guard it seemed perfectly legit. But now we know better!