Measuring the Earth Like Eratosthenes — a 2000 Year Old Experiment With a 3D-Printed Tool

Ancient History (pre-400)

Over 2,200 years ago, Eratosthenes calculated the circumference of the Earth using nothing more than shadows and geometry.

No satellites, no modern instruments, just careful observation.

I wanted to test that method myself, not by trusting known values, but by measuring the Earth from scratch.

So I built a standardized 3D printed measurement tool, coordinated a group of participants across different locations, and recreated this experiment during the March Equinox.

Each participant measured the Sun’s angle at solar noon using the same setup. We then combined the data to estimate Earth’s circumference.

The result came within about 0.3 percent of the accepted value.

Place your YouTube video here. This gives immediate proof and helps the project stand out.

Historical Context (Ancient History Category)

This project is based on the work of Eratosthenes, a Greek mathematician who lived from 276 to 194 BCE.

He observed that in one city the Sun cast no shadow at noon, while in another it did. From this difference, he calculated the size of the Earth with impressive accuracy.

This project follows the same idea using modern coordination, standardized tools, and real data.

The method is ancient, but the execution is modern.

What This Project Shows

This project turns a simple measurement into a large scale result.

You will see that the Sun’s angle changes depending on latitude. Different locations produce different shadow lengths. Those differences reveal the curvature of the Earth.

The key idea is that a very large system can be measured using small, simple observations.

How It Works

At solar noon the Sun reaches its highest point in the sky.

If the Earth is curved, people at different latitudes will measure different Sun angles.

The shadow length and pole height allow you to calculate that angle.

θ = arctan(shadow length / pole height)

Once you compare two locations, you can scale that difference to the full 360 degrees of the Earth.

Circumference = (distance ÷ angle difference) × 360

Watch the Full Experiment and breakdown here

https://www.youtube.com/watch?v=uT025d0bMmA

Measurement Setup

1. Vertical stick or rod
2. Flat surface
3. Measuring device (ruler or printed base)

My Standardized Tool (Optional but Recommended)

1. 3D printed base
2. Modular printed pole OR 1/2" wooden dowel

Other

1. Smartphone
2. Solar noon reference from timeanddate.com
3. Google Form for collecting data
4. Spreadsheet software

Goal

Create a reliable and repeatable way to measure shadow length as accurately as possible.

The entire experiment depends on this measurement. Small errors here will directly affect your final result.

Option A: Minimal Setup

You can perform this experiment with very basic tools.

What you need:

1. A straight stick or rod
2. A flat surface or ground
3. A ruler or measuring tape

How to use it:

Place the stick vertically and measure the length of the shadow at solar noon.

When this works well:

1. Quick test runs
2. Small scale experiments
3. No access to a 3D printer

Limitations:

1. Each participant may set things up differently
2. Small inconsistencies can introduce measurement error
3. Harder to compare results between people

Option B: Standardized Measurement Tool (Recommended)

For this experiment, I designed a 3D printable tool to reduce variation between participants and improve consistency.

This becomes especially important when combining data from multiple locations.

Design Overview

The tool has three main parts:

1. A vertical pole that casts the shadow
2. A horizontal base that captures the measurement
3. Markings spaced in 5 millimeter increments

This allows every participant to measure in the same way.

Why Standardization Matters

When multiple people are involved:

Different setups lead to different sources of error

Those errors compound when data is combined

A standardized tool ensures:

1. Consistent geometry
2. Comparable measurements
3. More reliable final results

How I Designed It in Tinkercad

The tool was designed to be easy to print, assemble, and modify.

Modular Pole System

The pole is split into multiple segments that screw together.

Why this matters:

1. Taller poles produce longer shadows, which improves measurement resolution
2. Smaller segments fit on most 3D printers
3. You can adjust height depending on your setup

Base Design

The base extends outward from the pole and acts as the measurement surface.

It includes evenly spaced markings for measuring shadow length.

Dovetail Connection System

The base is split into sections using dovetail joints.

This allows:

1. Larger measurement area without needing a large print bed
2. Clean alignment between parts
3. Easy assembly without additional hardware

This was created in Tinkercad using simple subtraction cuts to form interlocking shapes.

Measurement System Evolution (Important)

This tool went through a critical design change.

Original Design

The first version used angle markings directly on the base.

The idea was to read the Sun’s angle directly from where the shadow landed.

Problem

The shadow does not originate from the center of the pole.

It originates from the edge.

This introduced a small but consistent angular error of about 0.3 degrees.

Updated Design

The tool was redesigned to measure linear distance instead of angle.

Now:

1. The base uses 5 millimeter spacing
2. You measure shadow length directly
3. The angle is calculated afterward

This approach is more accurate and works with either a printed pole or a simple wooden dowel.

Print Failure and Fix (Engineering Detail)

During printing, the original pole design failed due to an internal overhang.

The printer could not support the geometry, which caused poor layer adhesion and failed prints.

Solution

The internal structure of the pole was redesigned with a gradual slope.

This removes unsupported overhangs and allows the part to print cleanly without supports.

Benefits:

1. More reliable prints
2. Stronger structure
3. Faster printing

Build Tips

1. Make sure the pole is perfectly vertical
2. Use a flat and level surface
3. Measure the pole height carefully and record it
4. Avoid flexible or bending materials
5. Take a test measurement before your actual run

the Design

https://makerworld.com/en/models/2119052-modular-eratosthenes-shadow-measurement-apparatus#profileId-3008666

Goal

Set up a system so multiple people can take consistent measurements that can be combined into a single result.

This is what turns a simple shadow measurement into a large scale experiment.

Why Coordination Matters

This experiment only works when:

1. Measurements are taken at the correct time
2. Data is collected in a consistent format
3. Locations are properly distributed

Without coordination, the data becomes difficult to compare or unreliable.

How I Organized It

I created a simple system using:

1. A Google Form for data submission
2. A shared spreadsheet for organizing results

Each participant submitted:

1. Their latitude
2. Shadow length
3. Pole height
4. Calculated angle or raw measurement
5. A photo of their setup

Choosing Participants

The most important factor is latitude spread.

You want participants:

1. As far apart as possible
2. Ideally aligned north to south

Why this matters:

1. Larger separation produces larger angle differences
2. Larger angle differences reduce percentage error

Number of Participants

Minimum: 2 people

Better: 5 or more

What I used: 9 participants

More data allows:

1. Cross checking
2. Outlier detection
3. More accurate averaging

Instructions Given to Participants

Each person was told to:

1. Use the same measurement method
2. Measure at solar noon
3. Record data carefully
4. Take a photo of their setup

Consistency across participants is more important than precision from any one person.

Data Structure

Keep your data clean and consistent.

Example fields:

1. Latitude
2. Shadow length
3. Pole height
4. Calculated angle
5. Notes or issues

Key Lesson

This step is what transforms the project from:

A simple demonstration

into

A coordinated scientific measurement system

Goal

Understand exactly what you are measuring and why this method works.

Without this, the process becomes mechanical. With it, the results become meaningful.

Core Idea

The experiment relies on one key observation:

The angle of sunlight changes depending on where you are on Earth.

If the Earth is curved:

1. People at different latitudes will measure different Sun angles
2. The difference between those angles reflects Earth’s curvature

What You Need

To make this work, you need three things:

1. At least two locations
2. Measurements taken at the same solar time
3. A way to determine distance between those locations

What You Are Actually Measuring

You are not directly measuring the Earth.

You are measuring:

1. The angle of sunlight at your location

Then comparing it with someone else’s measurement.

That difference represents how much the Earth curves between you.

Why This Works

If you know:

1. How much the Earth curves over a known distance

You can scale that up to a full 360 degree circle.

This is exactly what Eratosthenes did.

Key Insight

The larger the distance between participants:

1. The larger the angle difference
2. The more accurate your final result

Goal

Ensure every participant measures the Sun at the same point in the sky.

This is one of the most critical parts of the experiment.

What Is Solar Noon

Solar noon is the moment when the Sun reaches its highest point in the sky at your location.

This is not always 12:00 on the clock.

How to Find It

Use dateandtime.com:

1. Enter your exact location
2. Look up solar noon for your date
3. Record the exact time

Why Timing Matters

If measurements are not taken at solar noon:

1. The Sun’s angle will be incorrect
2. Results will not match between participants
3. Final calculations can become meaningless

Real Example From This Experiment

One participant measured about 30 minutes early.

Result:

1. The angle was completely incorrect
2. It created impossible calculations
3. That data had to be removed

Best Practices

1. Set a timer at least 10 minutes before solar noon
2. Be ready and set up early
3. Double check your location settings
4. Take the measurement as close to the exact time as possible

Goal

Capture an accurate and verifiable shadow measurement.

This is your primary data point.

Setup

1. Place the tool on a flat, stable surface
2. Ensure the pole is perfectly vertical
3. Confirm your measurement markings are visible

Measurement Process

At solar noon:

1. Observe the shadow cast by the pole
2. Mark the exact tip of the shadow
3. Measure the distance from the base of the pole to the shadow tip
4. Record the value immediately

Documentation (Very Important)

Each participant should:

1. Take a clear photo of the setup
2. Capture the shadow tip and measurement markings
3. Record their latitude and time

Why Photos Matter

Photos serve as:

1. Verification of correct setup
2. Evidence for your results
3. A way to catch mistakes later

Common Sources of Error

1. Pole not perfectly vertical
2. Uneven ground
3. Blurry or unclear shadow edge
4. Measuring from the wrong reference point

Tips for Accuracy

1. Use a sharp shadow edge, not a fuzzy one
2. Take multiple measurements and compare
3. Avoid windy conditions
4. Recheck your measurement before submitting

Here is all the shareable Data from the experiment

https://docs.google.com/spreadsheets/d/1El6jSTKcbkNlm7AMYMXTPXjaLBvDne5bMbN7Ph50vZc/edit?gid=0#gid=0

Goal

Turn your physical measurement into a usable angle.

This is where raw data becomes meaningful.

What You Measured

You now have:

1. Shadow length
2. Pole height

What You Need

The angle of the Sun relative to the ground.

The Relationship

The shadow and pole form a right triangle.

This allows you to calculate the angle using trigonometry.

θ = arctan(shadow length / pole height)

How to Calculate It

You can use:

1. A calculator with arctan
2. Spreadsheet software
3. Online tools

Important Details

1. Use consistent units for both values
2. Double check your numbers before calculating
3. Small errors here will affect your final result

Why This Step Matters

This angle is the core measurement used to:

1. Compare locations
2. Determine curvature
3. Calculate Earth’s size

Goal

Measure how far apart your data points are.

This connects your angle measurement to real-world scale.

What You Need

Distance between two participants, ideally aligned north to south.

Best Method: Real Distance Measurement

Have one participant travel to the other location and record distance using:

1. Car odometer
2. GPS tracking

Why This Is Preferred

1. Uses real-world measurement
2. Avoids map projection errors
3. Increases overall accuracy

Alternative Method

Use latitude coordinates:

1. Find difference in latitude
2. Convert degrees into distance

This is less direct but still usable.

Key Insight

Accuracy improves when:

1. Locations are farther apart
2. Alignment is closer to north and south

Goal

Scale your measured segment to the full size of the Earth.

What You Have

1. Distance between two locations
2. Difference in Sun angle

The Formula

Circumference = (distance ÷ angle difference) × 360

What This Means

You are asking:

If this small segment represents part of a circle, how big is the full circle?

Best Practices

1. Use multiple pairs of participants
2. Focus on pairs with large separation
3. Avoid using questionable data

Why Multiple Calculations Help

Different pairs will produce slightly different results.

Averaging them:

1. Reduces random error
2. Improves confidence in your result

Goal

Turn individual measurements into a reliable dataset.

How Data Was Collected

1. 9 real participants
2. Data submitted through a Google Form
3. Automatically compiled into a spreadsheet

What Was Recorded

1. Latitude
2. Shadow measurement
3. Calculated angle
4. Photo evidence

Observed Pattern

As latitude increased:

1. The measured Sun angle decreased

This matches the expected behavior on a curved Earth.

Outlier Detection

One measurement showed a major inconsistency.

Cause:

1. Measurement taken at the wrong time

Effect:

1. Produced impossible results

Action Taken

1. The data point was removed from calculations

Why This Matters

Removing bad data is not cheating.

It is necessary for:

1. Accuracy
2. Reliability
3. Scientific integrity

Goal

Show exactly how raw data becomes a final result.

Selected Data Points

Participant A:

1. Latitude: 32.5
2. Angle: 57.70

Participant B:

1. Latitude: 45.1
2. Angle: 45.20

Step 1: Latitude Difference

45.1 − 32.5 = 12.6 degrees

Step 2: Convert to Distance

12.6 × 68.95 = 866 miles

Step 3: Angle Difference

57.70 − 45.20 = 12.5 degrees

Step 4: Apply Formula

(866 ÷ 12.5) × 360 = 24,941 miles

Result

This is within about 0.33 percent of the accepted value.

Why This Is Powerful

This result comes from:

1. Simple tools
2. Real measurements
3. A method over 2,000 years old

Goal

Understand what your results mean.

Final Results

1. Best result: about 0.33 percent error
2. Average result: about 1.94 percent error

Accepted Value

Approximately 24,860 to 24,901 miles

What This Confirms

1. The method works
2. The measurements are valid
3. The Earth’s curvature is measurable using simple tools

Key Insight

Accuracy improves when:

1. Measurements are precise
2. Timing is correct
3. Distance is well measured

Goal

Make the experiment accessible. and the best would be two people that could drive almost perfecty north to south from eachother atleast 200 miles

Minimum Setup

1. Two participants
2. Two locations
3. Measurement tools
4. Known distance

Process

1. Measure at solar noon
2. Record shadow and height
3. Calculate angles
4. Compare results
5. Apply the formula

Recommendation

Choose locations that are:

1. Far apart
2. Roughly aligned north to south

Timing Errors

Problem: Measurements taken too early or late

Fix: Use precise solar noon timing

Measurement Errors

Problem: Inconsistent setups

Fix: Use a standardized tool

Design Issues

Problem: Tool inaccuracies or print failures

Fix: Iterate and refine design

Distance Errors

Problem: Inaccurate location spacing

Fix: Use real-world travel measurements

Ways to Expand This Project

1. Increase number of participants
2. Collect global data
3. Automate angle calculations
4. Build a dedicated app
5. Improve tool precision

Advanced Ideas

1. Use digital sensors instead of manual measurement
2. Compare results across seasons
3. Analyze error sources more deeply
4.