# Earth Curvature Calculator

Created by Davide Borchia
Last updated: Jul 12, 2022

If you have ever wondered how far can you see, the Earth curvature calculator will give you the answer. Keep reading this article: we will guide you on a conversational tour around the globe... curvature.

Here you will find:

• A quick clarification: Earth. Is. Round.
• How to calculate the distance to your horizon as an effect of the curvature of the Earth;
• Why boats disappear: the curvature of the Earth and the calculation of the hidden height of an object.

## Making things clear: is the Earth round?

Yes.

That's it!

Seriously speaking, around us, there's plenty of evidence of an almost-spherical planet, from the day-night cycle to the variation of the seasons with the latitude (get on a plane and try it yourself 😉), from geodesic routes to the disappearance of objects below the horizon. We even sent satellites in Earth's orbit!

Humans have known that the Earth 🌍 is round since the time of the Ancient Greeks. From there on, almost no one believed in a flat Earth. With the arrival of the Internet and social media, baseless pseudoscientific theories resurfaced and gained an audience. These are, however, nothing but a resonating chamber for a small number of individuals who gain notoriety thanks to their controversialism: the less we talk about them, the better for true science.

🙋 Earth is not round! Calm down before immolating our website on the pyre of science. The real shape of our planet is called the geoid: only with the development of satellites, we have had the possibility of measuring it accurately.

And after all, we even have pictures: thought-inspiring snapshots of our fragile world from the darkness of space. The Earth rising over the Moon's horizon as seen from the command module of Apollo 8. (credits: NASA)

Instead of doubting the shape of Earth, we should come together to defend this .

## What are the effects of the curvature of the Earth on the line of sight?

The existence of Earth's curvature is the reason we have a horizon. The horizon is the imaginary line after which an object lying on the ground would be covered by Earth's curvature, effectively entering a "shadow" area.

Which parameters affect the horizon? We can pinpoint, for example:

• The topography of the region surrounding you;
• The height of the observer's eyes;
• The atmospheric conditions.

Obviously, a mountain range ⛰ five kilometers away would reduce your horizon by a lot. However, it's hard (read: impossible) to calculate this: we can only consider the height of the observer's eyes in our Earth's curvature calculations.

It's immediate that the higher your observation point this, the farther you can see: it's no surprise to anyone that from a mountain summit, you can see, for example, a city that usually lies beyond the horizon.

## How do you calculate the distance to the horizon?

To calculate the distance to the horizon, we only need to know the height of your eyes above the ground.

Technically, you should use the height above sea level. However, this input would give you meaningful results only in the case of a perfectly spherical Earth. If you were on a plateau, it's probably better to insert your height relative to the height of the terrain you're on at the moment.

This difference becomes minimal the higher you are.

Let's do the math: you will be surprised by its simplicity.

The formula to calculate the effect of the curvature of the Earth on how far can you see is:

$\footnotesize d = \sqrt{(R_{\text{E}}+h)^2 - R_{\text{E}}^2}$

Where:

• $d$ is the calculated distance of the horizon;
• $R_{\text{E}}$ is the Earth's radius ($6,371,000\ \text{m}$); and
• $h$ is the observer's height.

As you can see, the effect of the curvature of Earth is calculated with nothing but a simple application of the Pythagorean theorem!

Imagine being on the beach 🏖. Assuming that you are THE average human, your eyes will lie at $161\ \text{cm}$ above the ground. Let's calculate the effect of the curvature of Earth on the line of sight:

\footnotesize \begin{align*} d &= \sqrt{(6,\!371,\!000+1.61)^2 - 6,\!371,\!000^2}\\ &= 4,\!530\ \text{m} \end{align*}

Your horizon would lie just shy of four kilometers and a half. Every object farther than the distance $d$, lying on the surface of the water, would be covered and not visible.

The question is spontaneous: can you calculate the curvature of Earth's effect on objects rising high above the ground?

## Calculate the height of the obscured portion of an object due to Earth's curvature

You are still on the beach (lucky!). A ship is approaching: what do you see? First, the topmost part of the vessel: antennas, chimneys, sails. Then, slowly, the rest of the boat will appear above the horizon: this effect is caused by Earth's curvature. The horizon obscures part of the boat, and only getting closer, the lower parts transit above that line.

🙋 This phenomenon is one of the best ways to experience firsthand the curvature of Earth: no one can dismiss it, Right? Right? And no, the boat is not sinking: buoyancy still exists!

The gradual disappearance of an object below the horizon thanks to the curvature of Earth is calculated using, again, the Pythagorean theorem.

Define the distance between you and the object, $a$. We built a right triangle subtracting the distance between you and the horizon $d$ from $a$: that's the distance between your horizon and the object.

Now we can calculate the height of the object that would have horizon at a distance $a-d$, with a common point between it and you:

\footnotesize \begin{align*} h_\text{hidden} &= \sqrt{(a-d)^2+R_{\text{E}}^2}\\ &=\sqrt{a^2-2\cdot a \cdot d +d^2 +R_{\text{E}}^2} \end{align*}

This is the height of the hidden portion of an object at distance $a$ from the observer.

Move from the sea to the mountains. Not so close, though: we are in Anchorage, Alaska. Go to , and look north. You would be at a distance of about $210\ \text{km}$ from Denali, the highest mountain in North America, with a height of $6,190\ \text{m}$. You would be again at sea level, so your eyes would lie at $165\ \text{cm}$ of height, and your horizon would be $d = 4,530\ \text{m}$. How much of Denali would be covered by Earth's curvature?

\footnotesize \begin{align*} h_\text{hidden} &=\left((2.10\times10^{5}-4,\!530)^2\right.\\ &\left.+6,\!371,\!000^2\right)=\\ &=3,\!310\ \text{m} \end{align*}

Only slightly more than half of the mountain would be covered"! Now take a car (or a bike!) and move toward Denali: you will slowly see the mountain rise in all its majesty.

🙋 A disclaimer: the results of our calculation may not be totally accurate. But worry not: it's not because of a lack of science: we simply neglected the effect of the air density and temperature on the refractive index of the atmosphere!

Davide Borchia
Distance to the object
mi
Eyesight level
ft
Distance to horizon
mi
Obscured object part
ft
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