Calculating the Difference Between Your Rooftop View and the View From the ISS
Pythagoras has a roof party.
Do you enjoy getting high?
A hypothetical construction project would see a new building added to the Tokyo skyline by 2045: a mile-high skyscraper, more than twice the height of the current tallest building in the world.
It sounds mighty grand, but such projects are invariably fraught with financial woes and elevator problems. We’ll believe in the Tokyo behemoth when we see it or, better yet, when we’re standing on top of it. Why our roof deck enthusiasm? Well, space travel is expensive, but trigonometry says that views from that high can be almost as epic as views from the stratosphere.
So, let’s talk about balls, in general, and Earth, in particular. When we stand at the top of a tall structure and look out to the horizon, we’re seeing some of the curvature of our spherical planet, as well. In order to calculate how far off that distant, hazy horizon is, we just have to understand the geometric nature of the our query and solve for X.
Before we do so, let’s go through the approximations that will make the math useful. Our planet is hardly a perfect sphere; it’s slightly oblong and studded with mountains and valleys, but a working figure for our planet’s radius — the “as-the-crow-flies” distance from sea level to the center of the Earth — is 6,378,100 meters. That figure comes from NASA.
The math we’ll be doing assumes that figure as the Earth’s radius, and assumes that the building you’re standing on top of is built at sea level. We’re assuming New York or Tokyo, not Denver, which is way more complex. Using the time-honored calculations of a guy named Pythagoras, we’re going to express this problem in terms of triangles. We already know the length of two sides of the triangle: one side is the radius of the Earth, the other side is that same radius plus the height of a building. Pythagoras famously demonstrated that a²+b²=c², so to find the length of this missing side of the triangle, we add the two squared figures together, then take a square root. The result is the distance to the horizon from your high-altitude vantage point.
How do we know this is a right triangle, because our line of site is, by definition, tangential to the Earth. The math from there is incredibly easy.
The Eiffel Tower is 984 feet tall, granting you about 38.4 miles of sight. The Empire State Building’s roof is at 1,250 feet above the ground. If you were to punch through the security guards and step upon it for the sake of the view, you’d see just over 43 miles away. A mile-high tower would offer a view of 89 miles.
There’s, unfortunately, no easy mental formula to turn a building’s number of floors into a sight distance, because we’re taking square roots here, and that gets complicated without a calculator rather quickly. In the interest of giving you some landmark figures to work with on the assumption that one story of a building equals ten feet in altitude, however, we present you with the following cheat sheet.
Five stories: 8.7 miles
Ten stories: 12.3 miles
15 stories: 15 miles
20 stories: 17.3 miles
25 stories: 19.4 miles
30 stories: 21.2 miles
40 stories: 24.5 miles
50 stories: 27.4 miles
60 stories: 30 miles
70 stories: 32.4 miles
80 stories: 34.7 miles
90 stories: 36.8 miles
100 stories: 38.7 miles
Depending on how invested you are in observing the curvature of the earth, it may behoove you to invest in an oxygen system for climbing Everest. Its summit is 29,029 high. You could see more than 208 miles away. To put that in perspective, crew members of the ISS can see a patch of the Earth with a diameter of roughly 2,000 miles at any given moment. That means that even the view from a mile-high skyscraper would only be slightly less than 0.8 percent of the size of the view from the ISS.
Keep training for liftoff.
