Earth Circumference Calculator (Using Sticks)
Recreate the ancient experiment of Eratosthenes to measure our planet by calculating the circumference of the Earth using sticks and shadows.
What is Calculating the Circumference of the Earth Using Sticks?
Calculating the circumference of the Earth using sticks is a method based on the famous experiment conducted by the Greek astronomer and mathematician Eratosthenes over 2,200 years ago. It is a foundational application of geometry and observation to determine the size of our planet. The core idea is that the Sun’s rays are essentially parallel when they reach Earth. By measuring the different shadow lengths cast by identical vertical sticks at two different locations (a known north-south distance apart) at the same time, one can calculate the curvature of the Earth and, from that, its total circumference.
This calculator is for anyone interested in astronomy, history of science, or hands-on educational projects. It helps you understand the genius of Eratosthenes’ simple method by allowing you to input your own (or historical) measurements. A common misunderstanding is that you need complex tools; in reality, as Eratosthenes showed, all you need is a stick, your feet (for measuring distance), and your brain!
The Eratosthenes Formula and Explanation
The logic behind calculating the circumference of the Earth using sticks is surprisingly straightforward. The angle of the sun’s shadow at one location directly corresponds to the angle separating the two measurement locations at the Earth’s center.
The primary formula is:
Circumference = (Distance × 360°) / Sun Angle (θ)
The Sun Angle (θ) itself is derived from your stick and shadow measurements using trigonometry:
Sun Angle (θ) = arctan(Shadow Length / Stick Height)
| Variable | Meaning | Unit (auto-inferred) | Typical Range |
|---|---|---|---|
| Stick Height | The vertical height of your gnomon. | meters / feet | 0.5 – 2.0 |
| Shadow Length | Length of the shadow at solar noon. | meters / feet | 0.0 – 1.0 (depends on latitude) |
| Distance | The North-South distance between measurement sites. | kilometers / miles | 500 – 1000 |
| Sun Angle (θ) | The calculated angle of the sun based on the shadow. | degrees | 1° – 15° |
Practical Examples
Example 1: Eratosthenes’ Historical Data
Eratosthenes knew that in Syene, the sun cast no shadow at noon on the summer solstice. In Alexandria, about 800 km to the north, a shadow was cast. He measured the angle to be about 7.2 degrees.
- Inputs:
- Stick Height: 1 m (hypothetical, as only the ratio matters)
- Shadow Length: ~0.126 m (This gives an angle of tan⁻¹(0.126/1) ≈ 7.2°)
- Distance Between Locations: 800 km
- Results:
- Sun’s Angle: 7.2°
- Calculated Circumference: (800 km * 360) / 7.2° = 40,000 km
Example 2: A Modern Amateur Experiment
Imagine two schools collaborate. One is in City A, and another is 650 miles directly north in City B. At solar noon, they measure shadows from a 3-foot stick. The school in City B measures a shadow of 0.47 feet. The school in City A is near the tropic and has almost no shadow.
- Inputs:
- Stick Height: 3 ft
- Shadow Length: 0.47 ft
- Distance Between Locations: 650 miles
- Results:
- Sun’s Angle: arctan(0.47 / 3) ≈ 8.9°
- Calculated Circumference: (650 miles * 360) / 8.9° ≈ 26,292 miles
Curious about solar noon? You might want to use a solar noon calculator to time your experiment perfectly.
How to Use This Earth Circumference Calculator
Follow these simple steps to perform your own calculation.
- Select Units: Choose between Metric (m/km) and Imperial (ft/mi) systems. The labels will update automatically.
- Enter Stick Height: Input the height of the vertical object you are using to cast a shadow.
- Enter Shadow Length: At solar noon, precisely measure the length of the shadow cast by your stick. This measurement should be from the location with the shadow (the northernmost location if one site has no shadow).
- Enter Distance: Input the north-south distance between the two locations where measurements are taken.
- Interpret Results: The calculator instantly provides the Earth’s circumference, the sun’s angle, and the implied radius and diameter. The chart visualizes how small the measured angle is compared to the full 360 degrees of the planet. For more on angles, check our angle calculator.
Key Factors That Affect the Calculation
- Vertical Stick: The stick (gnomon) must be perfectly perpendicular to the ground. Any tilt will alter the shadow length and skew the result.
- Solar Noon Timing: The shadow measurement must be taken at the exact moment of local solar noon, when the sun is at its highest point and the shadow is at its shortest.
- North-South Alignment: The accuracy of the Eratosthenes experiment calculator depends heavily on the two locations being on or very near the same line of longitude (a direct north-south line).
- Distance Measurement Accuracy: An accurate measurement of the distance between the two cities is critical. Eratosthenes was famously rumored to have hired bematists (trained walkers) to pace out the distance.
- A Perfectly Spherical Earth: The calculation assumes Earth is a perfect sphere. In reality, it is an oblate spheroid (slightly flattened at the poles), which introduces a small error. The modern accepted circumference is about 40,075 km.
- Parallel Sun Rays: The model relies on the sun being so far away that its rays arrive at Earth in parallel lines. This is an excellent and valid approximation.
Frequently Asked Questions (FAQ)
1. Why do I need two different locations?
The entire method is based on the *difference* in the sun’s angle between two points. If you only measure at one point, you can find your latitude, but you cannot determine the planet’s size. Check out our guide on the DIY geodesy for more info.
2. Does the height of the stick matter?
No, the absolute height doesn’t matter, but it must be known accurately. The calculation depends on the *ratio* of the shadow length to the stick height. A taller stick will produce a longer shadow, but the angle derived will be the same.
3. What if neither location has a zero-length shadow?
You can still do the experiment! You would measure the shadow angle at both locations and use the *difference* between the two angles as your “Sun Angle” in the formula. For example, if you measure 7° in one city and 9° in another, the angle to use is 2°.
4. Why must the measurement be at solar noon?
Solar noon is when the sun crosses the local meridian. At this point, a shadow cast by a vertical stick will point directly north or south, simplifying the geometry and ensuring you are measuring the correct angle related to latitude.
5. How accurate was Eratosthenes’ measurement?
His measurement was remarkably accurate, estimated to be within 2-15% of the actual value. The uncertainty comes from the fact that we don’t know the exact length of the “stadion,” the unit of measurement he used.
6. Can I use this calculator with any units?
Yes. Our calculator supports both metric and imperial units. You can perform conversions using a tool like our distance converter, but the calculator handles it for you as long as you select the correct system.
7. What does the pie chart represent?
The pie chart shows a visual representation of the sun’s angle you calculated (the small slice) in proportion to the full 360 degrees of the Earth’s circumference (the rest of the circle).
8. What is the best way to measure the distance between two cities?
Today, you can easily use online mapping services to get a very precise north-south distance (meridional distance) between two points.
Related Tools and Internal Resources
If you found this tool for calculating the circumference of the earth using sticks useful, you might also enjoy these related resources:
- Solar Noon Calculator: Find the exact time for your measurement.
- Angle Calculator: A tool for exploring trigonometric functions like arctan.
- Distance Converter: Easily switch between kilometers, miles, and other units.
- Scientific Notation Calculator: Useful for handling very large numbers like planetary measurements.
- Guide to Latitude and Longitude: Understand the coordinate system that makes this experiment possible.
- The History of Astronomy: Learn more about figures like Eratosthenes and their groundbreaking discoveries.