Calculate Earth’s Radius Using Shadows (Eratosthenes Method)

An interactive tool to estimate Earth’s circumference and radius based on simple shadow measurements.

Eratosthenes Calculator

Enter your measurements from two different locations (or two different times in the same location) to calculate the Earth’s radius.

Results copied!

Calculation Results

Formula Used:
The Earth’s circumference (C) is proportional to the angle difference (Δθ) between the two locations, relative to the full circle (360 degrees). The radius (R) is calculated from the circumference using C = 2 * π * R.

Mathematically:
(Distance / Δθ) = (Circumference / 360°)
Circumference = Distance * (360° / Δθ)
Radius = Circumference / (2 * π)

What is the Eratosthenes Experiment for Calculating Earth’s Radius?

The experiment to calculate the radius of the Earth using shadows is a brilliant and historically significant method, most famously demonstrated by the ancient Greek mathematician and geographer Eratosthenes of Cyrene around 240 BC. It’s often referred to as Eratosthenes’ experiment. This method relies on simple geometry and the observation of the sun’s position at different locations on Earth simultaneously. It’s a foundational example of how scientific inquiry and basic measurements can lead to profound discoveries about our planet.

Who Should Use It?

• Students and Educators: Perfect for demonstrating principles of geometry, trigonometry, astronomy, and scientific method.
• Science Enthusiasts: Anyone curious about how we know the Earth is round and its approximate size without modern technology.
• Amateur Astronomers/Geographers: Individuals interested in historical scientific achievements and practical applications of geometry.

Common Misconceptions:

• Requires Complex Equipment: The original experiment used a simple vertical stick (gnomon) and observations. While modern tools can improve accuracy, the core concept is simple.
• Only Works at Equinoxes/Solstices: While Eratosthenes used the summer solstice, the principle can be applied at any time of year, provided the sun’s angles are measured accurately at both locations simultaneously (or at the same local time, like solar noon).
• Assumes Flat Earth Locally: The experiment relies on the understanding that the Earth is curved, but for short distances, the ground can be approximated as flat. The difference in the sun’s angle is the key indicator of curvature.
• Needs to Be Done Far Apart: While a larger distance yields a more accurate result (as seen in the formula: Circumference = Distance * (360° / Δθ)), the principle works even with moderate distances.

Earth’s Radius Formula and Mathematical Explanation

Eratosthenes’ method is a beautiful application of geometry. Imagine the Earth as a sphere. The sun’s rays, coming from an extremely distant source, can be considered parallel when they reach Earth. If the Earth were flat, parallel rays would hit all vertical objects at the same angle. However, because the Earth is curved, the angle of these parallel rays relative to vertical objects will differ at different latitudes.

Eratosthenes made two key observations:

1. In Syene (modern Aswan, Egypt), on the summer solstice at noon, the sun was directly overhead. This meant vertical objects cast no shadow, and sunlight reached the bottom of deep wells. The angle of the sun’s rays relative to a vertical line was 0 degrees.
2. At the same time in Alexandria (a city directly north of Syene), vertical objects cast a measurable shadow. Eratosthenes measured the angle of this shadow.

He measured the angle of the shadow in Alexandria to be approximately 7.2 degrees. This 7.2-degree difference is the same as the angle subtended at the Earth’s center by the arc connecting Syene and Alexandria.

The Core Calculation:

The ratio of the distance between the two cities to the Earth’s total circumference is the same as the ratio of the angle difference between the two locations to the total degrees in a circle (360°).

Formula Derivation:

Let:

• D = Distance between the two locations (e.g., Syene and Alexandria)
• Δθ = The difference in the angle of the sun’s rays between the two locations (measured from the vertical). This is |angle2 - angle1|.
• C = Earth’s Circumference
• R = Earth’s Radius
• π = Pi (approximately 3.14159)

The proportion is:

D / C = Δθ / 360°

To find the Circumference (C):

C = D * (360° / Δθ)

To find the Radius (R):

We know that C = 2 * π * R.

Therefore, R = C / (2 * π).

Substituting the formula for C:

R = [ D * (360° / Δθ) ] / (2 * π)

Variables Table:

Variable	Meaning	Unit	Typical Range / Notes
Distance (D)	North-South distance between measurement points.	km (or miles)	> 100 km for reasonable accuracy. Eratosthenes used ~5000 stadia (~800 km).
Angle 1 (θ1)	Angle of sun’s rays from vertical at Location 1.	Degrees (°)	0° at local solar noon on solstice in specific locations; otherwise, small positive angle.
Angle 2 (θ2)	Angle of sun’s rays from vertical at Location 2.	Degrees (°)	Measured simultaneously or at local solar noon.
Angle Difference (Δθ)	Absolute difference between Angle 1 and Angle 2.	Degrees (°)	> 0.1° for reliable calculation. Must be less than 180°.
Circumference (C)	The distance around the Earth’s equator (or along a meridian).	km (or miles)	Approx. 40,075 km at the equator.
Radius (R)	Distance from the Earth’s center to its surface.	km (or miles)	Approx. 6,371 km (mean radius).
Pi (π)	Mathematical constant.	Unitless	~3.14159

Practical Examples of Calculating Earth’s Radius

Let’s walk through a couple of scenarios using the Eratosthenes method.

Example 1: Replicating Eratosthenes’ Findings

Imagine you are performing Eratosthenes’ experiment:

• Location 1 (Syene): At local solar noon on the summer solstice, a vertical stick casts no shadow. Angle 1 = 0°.
• Location 2 (Alexandria): Directly north of Syene, at the same time, a vertical stick casts a shadow. You measure the angle of the sun’s rays to be 7.2°. Angle 2 = 7.2°.
• Distance: The estimated distance between Syene and Alexandria is approximately 800 km.

Using the Calculator:

• Distance Between Locations: 800 km
• Angle of Shadow at Location 1: 0 degrees
• Angle of Shadow at Location 2: 7.2 degrees

Calculator Output:

Calculated Earth Radius: ~6369 km

Calculated Earth Circumference: ~40013 km

Difference in Angles: 7.2 degrees

Assumed Ratio: 111.11

Interpretation: This result is remarkably close to the modern accepted mean radius of the Earth (~6,371 km). Eratosthenes’ original calculation, using his estimates for distance and angle, yielded a circumference of about 250,000 stadia, which is estimated to be within 1-15% of the actual value, an incredible feat for his time.

Example 2: A Modern Experiment Across Continents

Let’s consider a more modern, hypothetical scenario:

• Location 1 (Chicago, USA): Measured at local solar noon. Angle of shadow = 15°.
• Location 2 (Paris, France): Measured at local solar noon. Angle of shadow = 10°.
• Distance: The great-circle distance between Chicago and Paris is approximately 6,150 km.

Using the Calculator:

• Distance Between Locations: 6150 km
• Angle of Shadow at Location 1: 15 degrees
• Angle of Shadow at Location 2: 10 degrees

Calculator Output:

Calculated Earth Radius: ~6258 km

Calculated Earth Circumference: ~39317 km

Difference in Angles: 5 degrees

Assumed Ratio: 123

Interpretation: This calculation gives a radius of about 6,258 km. While slightly different from the mean radius, this value is still within a reasonable margin, considering potential inaccuracies in measurements (e.g., exact local solar noon, precise angle measurement, non-perfectly parallel sun rays due to proximity, and Earth’s oblateness). This demonstrates the robustness of the Eratosthenes principle even with imperfect conditions.

How to Use This Earth Radius Calculator

Using this calculator is straightforward. Follow these steps to estimate the Earth’s radius based on the Eratosthenes method:

Step-by-Step Instructions:

1. Choose Two Locations: Select two distinct locations that are roughly aligned along a north-south axis. The further apart they are, the more accurate your result will likely be.
2. Measure Shadow Angles: At precisely local solar noon on the same day for both locations, measure the angle of the shadow cast by a vertical object (like a stick or pole of known height).
   • Location 1: Measure the angle of the sun’s rays relative to the vertical object. Enter this value in ‘Angle of Shadow at Location 1 (degrees)’. If the sun is directly overhead (no shadow), this angle is 0°.
   • Location 2: Measure the angle similarly. Enter this value in ‘Angle of Shadow at Location 2 (degrees)’.

   Accuracy is key here! Ensure your object is perfectly vertical and your angle measurement is precise.

3. Determine Distance: Find the approximate north-south distance between your two chosen locations. This can be done using mapping tools or geographical databases. Enter this distance in ‘Distance Between Locations (km)’.
4. Input Values: Carefully enter the measured ‘Distance’, ‘Angle 1’, and ‘Angle 2’ into the respective fields in the calculator.
5. Calculate: Click the ‘Calculate Radius’ button.

How to Read the Results:

• Calculated Earth Radius: This is the primary output, representing your estimated radius of the Earth in kilometers.
• Calculated Earth Circumference: Derived from your calculated radius, this shows the estimated total distance around the Earth.
• Difference in Angles: This is the absolute difference between the two shadow angles you entered (|Angle 2 - Angle 1|). It’s the crucial value representing the Earth’s curvature between your locations.
• Assumed Ratio: This is the factor (360 / Δθ) by which your measured distance is multiplied to estimate the full circumference.

Decision-Making Guidance:

While this calculator provides an estimate, the accuracy depends heavily on the precision of your inputs. Use the results to:

• Compare with Accepted Values: See how close your calculated radius is to the scientifically accepted mean radius of ~6,371 km.
• Understand Scientific Principles: Gain a practical understanding of geometry, spherical trigonometry, and the curvature of the Earth.
• Improve Measurements: If your results are far off, review your measurement techniques for angles and distance. Experimenting further can refine your understanding.

Key Factors Affecting Earth Radius Calculation Results

The accuracy of the Eratosthenes method, and thus the output of this calculator, is influenced by several factors. Understanding these is crucial for interpreting your results:

1. Accuracy of Angle Measurement: This is paramount. Even small errors in measuring the shadow angle (e.g., ±0.1°) can significantly impact the calculated radius, especially if the distance is large. Ensure the measuring stick is perfectly vertical and measurements are taken at the exact local solar noon.
2. Accuracy of Distance Measurement: The ‘Distance Between Locations’ input directly scales the calculated circumference. Errors in determining the precise north-south distance (great-circle distance) will lead to proportional errors in the final radius. Using reliable mapping tools is essential.
3. Simultaneity of Measurements: Ideally, angles should be measured at the exact same moment, or at least at local solar noon for both locations on the same day. If measurements are taken at different times of day or on different days, the sun’s angle will vary due to Earth’s rotation and orbit, introducing errors.
4. Geographical Alignment: The calculation assumes the two locations lie on the same meridian (a line of longitude). If there’s a significant east-west separation, the assumption of parallel sun rays hitting surfaces perpendicular to the local vertical becomes less precise for calculating latitudinal distance.
5. Earth’s Shape (Oblateness): The Earth is not a perfect sphere; it’s an oblate spheroid, bulging at the equator and flattened at the poles. This means the radius varies depending on latitude. The Eratosthenes method inherently calculates the radius along the meridian connecting the two points, which might differ slightly from the equatorial radius.
6. Atmospheric Refraction: The Earth’s atmosphere bends sunlight, especially near the horizon. While less significant at noon when the sun is high, it can slightly alter the apparent position of the sun and affect precise angle measurements.
7. Sun’s Apparent Diameter: The sun is not a point source of light; it has an angular diameter of about 0.5°. This means shadows have fuzzy edges, making precise measurement difficult. Eratosthenes cleverly used the fact that in Syene, the sun was directly overhead (no shadow), avoiding this issue for one location.
8. Surface Level Measurements: The calculation assumes measurements are taken at sea level. Altitude differences between locations can slightly affect the perceived angle of the sun.

Frequently Asked Questions (FAQ)

Can I use any two locations?

For the most accurate results, the two locations should be roughly aligned along a north-south axis (same longitude). If they are significantly east-west of each other, the distance measurement needs to account for the curvature of the Earth at different latitudes, and the ‘distance’ input should represent the north-south component of that separation.

What if I measure at different times of the day?

It’s crucial to measure the shadow angle at local solar noon for both locations on the same day. Local solar noon is when the sun reaches its highest point in the sky. Measuring at different times means the sun’s angle is affected by both the Earth’s curvature and its rotation, leading to inaccurate results for this specific calculation.

How accurate is the Eratosthenes method?

The accuracy depends heavily on the precision of the distance and angle measurements. Eratosthenes’ original calculation was remarkably close, estimated to be within 1-15% of the true value. With modern tools (GPS for distance, precise angle measuring devices), much higher accuracy is possible, but perfect results are challenging due to factors like atmospheric refraction and Earth’s non-spherical shape.

What if my angle difference is very small?

A small angle difference (e.g., less than 1 degree) is fine, but it requires a very large distance between the locations to yield a reliable circumference estimate. If the distance is also small, the resulting calculation might be highly inaccurate. The calculator requires a difference greater than 0.1 degrees to avoid division by a near-zero number.

Can I use this method at night?

No, this method relies on measuring the angle of the sun’s rays using shadows cast by sunlight. It cannot be performed at night.

What if the ground isn’t perfectly flat?

The angle measurement assumes the base for your vertical object is perfectly level. Minor unevenness might introduce small errors. For best results, ensure the ground where you place your measuring stick or transit is as flat and level as possible.

Does the height of the stick matter?

The height of the stick matters for measuring the shadow’s length, but for calculating the *angle* of the sun’s rays, it’s the ratio of shadow length to stick height (tangent of the angle) or direct angle measurement with a protractor/transit that’s important. A taller stick can provide a more easily measurable shadow, potentially increasing accuracy if measured carefully.

Why are my results different from the accepted Earth radius?

Several factors contribute: measurement inaccuracies (angles, distance), non-simultaneous measurements, non-perfect north-south alignment, Earth’s oblateness, atmospheric effects, and the sun not being a perfect point source. Even Eratosthenes’ results were approximations.

Tables & Charts

Sun Angle vs. Latitude Approximation

The chart above illustrates how the sun’s angle at local solar noon changes with latitude, assuming measurements are taken on the spring or autumn equinox (when the sun is directly over the equator). The relationship is linear if the Earth were a perfect sphere and the sun’s rays parallel.

Eratosthenes Experiment Data Summary

Scenario	Location 1 Angle (°)	Location 2 Angle (°)	Distance (km)	Angle Diff (°)	Calculated Radius (km)	Calculated Circumference (km)
Eratosthenes (Approx)	0.0	7.2	800	7.2	—	—
Hypothetical Chicago-Paris	15.0	10.0	6150	5.0	—	—

This table summarizes the inputs and calculated outputs for the practical examples discussed.

Related Tools and Resources

• Eratosthenes Earth Radius Calculator – Our interactive tool to perform the calculation.
• Celestial Navigation Basics – Learn how ancient mariners used stars and the sun.
• Spherical Geometry Explained – Deeper dive into the math behind our spherical world.
• History of Astronomy Milestones – Key discoveries and figures in understanding the cosmos.
• Latitude and Longitude Finder – Helps determine distances between locations.
• Field Angle Measurement Techniques – Practical guides for accurate measurements.
• Earth Science Fundamentals – Understand the physical properties of our planet.