Earth’s Circumference from a Sunset Calculator

A modern tool for recreating a classic physics experiment. Calculate our planet’s size by observing a sunrise or sunset from two different heights.

What is the ‘Calculate Earth’s Circumference Using Sunrise’ Method?

This method is a clever, modern adaptation of the principles used by ancient astronomers like Eratosthenes to measure our planet. Instead of measuring shadows in two different cities, this technique uses the time difference of a sunset (or sunrise) observed from two different altitudes at the same location. The core concept relies on the fact that an observer at a higher altitude can see “further” over the curve of the Earth, and will therefore see the sun for a slightly longer period as the planet rotates.

By precisely timing how much longer the sun is visible from a higher point (e.g., standing up) compared to a lower point (e.g., lying down), we can use geometry and the known rotation speed of the Earth to deduce its radius, and consequently, its circumference. This experiment is a favorite among students, amateur astronomers, and anyone curious about applying fundamental physics to understand the world around them. One common misunderstanding is that this yields a perfectly exact number; in reality, it’s an excellent approximation subject to factors like atmospheric refraction and timing accuracy.

The Formula and Explanation

The calculation hinges on the geometric relationship between the observer’s height, the Earth’s radius, and the angle the Earth rotates in the observed time difference. The formula to find the Earth’s radius (R) is:

R = 2 * (sqrt(h2) – sqrt(h1))² / (ω * Δt)²

Once the radius (R) is found, the circumference (C) is easily calculated with the classic formula C = 2 * π * R.

Formula Variables

Variable	Meaning	Unit	Typical Range
R	Radius of the Earth	km or miles	~6,371 km
h1	Lower observation altitude	meters or feet	0.1 – 2 m
h2	Higher observation altitude	meters or feet	1.5 – 100 m
Δt	Time difference between sunsets	seconds	5 – 20 s
ω (omega)	Earth’s angular velocity	radians/second	~7.27 x 10^-5 rad/s
π (pi)	The mathematical constant Pi	Unitless	~3.14159

Practical Examples

Example 1: A Person on a Beach

An observer at the beach decides to perform the experiment. They lie down so their eyes are 0.2 meters above the ground and start a timer when the last speck of sun disappears. They quickly stand up, bringing their eye level to 1.7 meters, and stop the timer when the sun disappears a second time.

• Inputs: h1 = 0.2 m, h2 = 1.7 m, Δt = 7.5 seconds
• Units: Metric
• Results: This would yield a calculated Earth circumference of approximately 40,200 km, an impressively close result!

Example 2: An Observer in a Tall Building

Someone watches the sunset from the ground floor of a building (eye level 2 meters). They then take a fast elevator to a balcony where their eye level is 100 meters high.

• Inputs: h1 = 2 ft, h2 = 100 ft, Δt = 33 seconds
• Units: Imperial
• Results: Using these inputs, the calculator would find a circumference of about 24,500 miles, again demonstrating the method’s viability. You can try a DIY earth circumference measurement at home.

How to Use This Calculator

1. Select Unit System: Choose ‘Metric’ for meters/km or ‘Imperial’ for feet/miles. The input labels will update automatically.
2. Enter Altitudes: Input your lower observation height (h1) and your higher observation height (h2). Be as accurate as possible.
3. Enter Time Difference: Input the time in seconds (Δt) you measured between the two sunsets (or sunrises).
4. Calculate: Click the “Calculate” button to see the results.
5. Interpret Results: The calculator will display the primary result (Earth’s Circumference) and intermediate values like the calculated radius. The chart will also update to show how your measurement compares to the official value. For more on this, see our guide on a modern Eratosthenes experiment.

Key Factors That Affect This Calculation

• Timing Accuracy: This is the most critical factor. Even a half-second error in measuring Δt can significantly alter the result. Using a digital stopwatch is essential.
• Altitude Precision: The accuracy of your height measurements for h1 and h2 directly impacts the formula.
• Atmospheric Refraction: The atmosphere bends light, making the sun appear higher than it is. This effect is strongest at the horizon and introduces a systematic error that makes the calculated circumference slightly larger than the true value.
• A Clear Horizon: This experiment works best with a perfectly flat, unobstructed horizon, such as the sea. Hills, buildings, or trees will interfere with the timing.
• Defining ‘Sunset’: It’s crucial to be consistent. Most experimenters time to the moment the very last gleam of the sun’s upper limb vanishes.
• Observer’s Latitude: The Earth is not a perfect sphere (it’s an oblate spheroid). The formula assumes a perfect sphere, so results will vary slightly depending on your location. Understanding the physics can be enhanced by reading about the sunset timing experiment.

Frequently Asked Questions (FAQ)

1. Why does this work for both sunrise and sunset?

The principle is the same. Whether the sun is disappearing (sunset) or appearing (sunrise), a person at a higher altitude will see it for longer or earlier, respectively. The time difference (Δt) is the key measurement in both cases.

2. How accurate can this calculator be?

With careful measurements, you can often get a result within 5-10% of the actual value. The biggest source of error is typically timing the sunset and atmospheric conditions.

3. Do I need a telescope?

No, a telescope is not necessary and can actually make it harder to judge the exact moment the sun’s limb crosses the horizon. The naked eye is sufficient.

4. Why do my results change when I switch units?

The calculator automatically converts all inputs to a standard internal unit (meters) for the formula, and then converts the final result back to your chosen system (kilometers or miles). This ensures the physics remains consistent. A guide to a similar experiment can be found here: how to measure Earth’s size.

5. What is the accepted circumference of the Earth?

The Earth’s equatorial circumference is officially about 40,075 km (or 24,901 miles). Our calculator uses this value in the comparison chart.

6. What’s a good, simple experiment to try?

Go to a beach or a large, flat field. Lie on your stomach (h1 ≈ 0.2m) and time the sunset. As soon as it disappears, stand up quickly (h2 ≈ 1.8m) and time the second sunset. You should get a time difference of 7-10 seconds.

7. Is there a similar historical experiment?

Yes, this is a modern take on the famous experiment by Eratosthenes, who used shadows in two different cities to calculate the circumference over 2,000 years ago.

8. Does my phone’s stopwatch work?

Yes, a smartphone stopwatch is an excellent and readily available tool for measuring the time difference with sufficient precision for this experiment.