Free Flat vs Round Earth Calculator

Introduction: The Flat vs Round Earth Calculator

Understanding the shape of our planet can be approached in two ways: through observation‑based verification or by ignoring evidence. Science, the method that took humanity to the Moon, relies on the former. The Flat vs Round Earth Calculator is a free online science experiment that lets you personally test whether the Earth is round using three simple yet powerful demonstrations. It serves as both an Earth curvature calculator and a hands‑on learning tool. By inputting a few measurements, you can simulate the Sunset Twice Experiment, the Disappearing Object Calculator, and the Stick Shadow Experiment (the classic Eratosthenes Earth circumference method).

The Flat‑Earth Belief and Its Shortcomings

Surveys indicate that roughly one in six people in the United States are uncertain about Earth's sphericity. This skepticism often links to a strong individualistic culture and a general distrust of institutions and experts. Flat‑Earth proponents frequently reject established scientific knowledge accumulated over two millennia.

Some have gone to extraordinary—and occasionally tragic—lengths to prove a flat disk rather than a sphere. “Mad Mike” Hughes built a steam‑powered rocket to photograph the horizon from 1,800 ft (550 m), unaware that curvature is only detectable above 35,000 ft (11 km). He died in a failed flight in 2020. YouTube personality Logan Paul planned a journey to Antarctica to find the “edge” of the Earth; the trip was canceled due to the pandemic. Rapper B.o.B tried to crowdfund a satellite launch but raised less than 1 % of his million‑dollar goal. All these expensive efforts could have been replaced by the low‑cost experiments described below.

The flat‑Earth model envisions the world as a disk with the North Pole at the center and a ring of ice around its circumference. The sun moves overhead at a much smaller distance than in reality, circling like a mobile. This arrangement cannot explain sunrises and sunsets (unless the disk spins like a coin, which would eliminate time zones). Computer simulations of such a model reveal that the sun never truly sets, its apparent size changes throughout the day, and seasons would not behave as observed—contradictions that the spherical model resolves effortlessly. Moreover, the disk model fails to account for many everyday observations: why are both the center (the Arctic) and the edge (Antarctica) cold? Why do aircraft routings over the southern hemisphere make sense only on a globe? These inconsistencies highlight the explanatory power of the spherical Earth model.

Sunset Twice: Seeing the Sun Again

On a spherical Earth, the setting sun’s light is blocked by the rising shadow of the horizon. If you move upward quickly enough, you can outpace that shadow and watch part of the sunset a second time. The time $t$ (in seconds) it takes the shadow to climb a height $h$ (in feet) is approximately

$$t = 8 \sqrt{h}.$$

The calculator then computes the percentage of a second sunset you would witness using:

$$\text{Sunset two percentage} = 100 \times \left(1 - \frac{s - t - t_{\text{travel}}}{s}\right),$$

where $s$ is the sunset duration at your location and $t_{\text{travel}}$ is the time you take to move from your low height to your high height.

The tool provides pre‑set scenarios to simplify input:

• Lie down, stand up – low height ~8 inches (20 cm), high height ~5 ft, travel time ~0.5 s. This yields about 5 % of a second sunset.
• Fast elevator in a skyscraper – modeled on the Burj Khalifa hotel: low height 0, high height 1,800 ft (550 m), travel time ~45 s. The calculator shows you can see 88 % of the sunset again.
• Drone with a camera – typical drone ascending to 3,300 ft (1,000 m) in 200 s; about one‑third of the sunset becomes visible.

You can also enter custom values for your own location and equipment. The sunset duration can be looked up for your latitude or measured directly. Simply specify the two eye levels, the time needed to change between them, and the local sunset length, and the Sunset Twice Experiment will instantly compute the fraction of a second sunset you can expect.

Disappearing Objects: Hiding Behind the Horizon

The curvature of the Earth hides the lower portion of distant objects when you lower your viewpoint. This effect is often summarized as 8 inches per mile (12 cm per kilometer), but an accurate calculation requires taking the observer’s height into account.

First, find the distance to the horizon from your viewpoint height $h$:

$$a = \sqrt{(r + h)^2 - r^2},$$

where $r$ is Earth’s radius (3,959 miles or 6,371 km). Next, the height of an object that is just obscured when viewed from ground level at a distance $d$ is:

$$x = \sqrt{a^2 - 2ad + d^2 + r^2} - r.$$

The difference between the obscured heights at a high and a low viewpoint tells you how tall an object can be made to disappear by changing your perspective.

To perform this experiment, choose a calm day and find a lake or stretch of sea with a far shore at least 2 km away. Use a camera on a tripod to measure two heights: standing up (high viewpoint) and as low as possible (e.g., lying down). The distance to the opposite shore can be measured using a paper map or the measuring tool in Google Maps. Enter the distance and the two heights into the Disappearing Object Calculator. The tool instantly shows how much of any target becomes hidden. If the Earth were flat, lowering your viewpoint would cause no change in how much of the object you see—yet in reality, the difference is unmistakable.

For a convincing demonstration, watch a truck disappear behind a lake: from a high viewpoint the entire truck is visible; after lowering the camera to near water level, the truck’s lower half—or the whole truck—vanishes. Offshore wind farms also illustrate the effect clearly: the farther a turbine, the more of its tower is concealed by the curvature.

Stick Shadows: Recreating Eratosthenes’ Measurement

Around 240 BC, the Greek scholar Eratosthenes used nothing but a stick, a shadow, and geometry to estimate Earth’s circumference with remarkable accuracy. This experiment not only proves Earth is round but also yields a direct measurement of its size.

At local noon on the same day, measure the shadow cast by a vertical stick at two locations separated north‑south by at least 100 miles (the greater the distance, the better). The angle $\theta$ between the stick and the sun’s rays is:

$$\theta = \arctan\left(\frac{\text{shadow length}}{\text{stick height}}\right).$$

Let $\theta_1$ and $\theta_2$ be the angles at the southern and northern sites, respectively. The angular separation $\alpha = \theta_2 - \theta_1$ corresponds to a fraction of the full 360° circle. If the north‑south distance between the two locations is $D$, Earth’s circumference is:

$$\text{Circumference} = D \times \frac{360^\circ}{\alpha}.$$

If you cannot travel, you can partner with a friend who lives far away. Ensure both sticks are the same length or measure their lengths individually, take readings at solar noon on the same day, and use the north‑south component of the distance between your sites. On a map, draw a line due north from the southern location and a line due east from the northern location; the distance from the southern location to the intersection is the required north‑south distance.

Example: A stick 3 ft (0.91 m) long casts a shadow of 3 ft at location A (shadow angle 45°) and 5.2 ft at location B (angle about 60°). The north‑south distance between A and B is 1,000 miles (1,609 km). The calculator returns an estimated circumference of 23,971 miles (38,577 km)—within 4 % of the true polar circumference of 24,860 miles (40,008 km).

If the Earth were flat, the same shadow data would force the sun to be only about 1,370 miles away (using the relation $d = h \times \left|\tan\theta_2 - \tan\theta_1\right|$), and its apparent diameter would change by 40 % over the same 1,000‑mile journey—something that is never observed. Thus the stick shadow experiment provides a direct, quantitative refutation of the flat‑Earth model.

For best results, perform the experiment on a sunny day near the summer or winter solstice to minimize seasonal variation in day length. The Flat vs Round Earth Calculator does all the heavy lifting—you only need to enter the stick length, shadow lengths, and travel distance to receive your own estimate of our planet’s circumference.