Earth Size Calculator: Applying Ancient Trigonometry

A modern tool for calculating the size of the Earth using trig, based on the historical method of Eratosthenes.

Eratosthenes’ Method Calculator

In-Depth Guide to Calculating the Size of the Earth Using Trig

What is Calculating the Size of the Earth Using Trig?

Calculating the size of the Earth using trig refers to the classical method pioneered by the Greek astronomer and mathematician Eratosthenes over 2,200 years ago. It is a brilliant application of basic geometry and observation to determine our planet’s circumference. The method doesn’t require advanced technology, only an understanding of angles, ratios, and the assumption that the Earth is a sphere and the sun’s rays are parallel. This earth circumference calculator is perfect for students, educators, and astronomy enthusiasts who want to replicate this historic experiment.

The Formula for Calculating Earth’s Size

The core of the calculation lies in a simple ratio. The ratio of the distance between two points on a meridian to the Earth’s entire circumference is equal to the ratio of the sun’s angular difference between those two points to the 360 degrees of a full circle. The primary formula is:

Earth Circumference = (Distance Between Points × 360) / Angle of Sun’s Rays (θ)

The angle θ is found using trigonometry from the shadow’s measurements:

θ = arctan(Shadow Length / Stick Height)

Variables in the Calculation

Variable	Meaning	Unit	Typical Range
Distance	The North-South distance between the two measurement locations.	km or mi	100 – 1,000
Stick Height	The height of the gnomon or vertical stick.	meters, feet	1 – 10
Shadow Length	The length of the shadow at local noon.	meters, feet	0 – 2 (highly dependent on latitude/time)
θ (Theta)	The angle of the sun’s rays, derived from the shadow.	Degrees	0 – 90

Practical Examples

Example 1: Eratosthenes’ Original Measurement

Eratosthenes knew the distance between Alexandria and Syene was about 5,000 stadia (approx. 800 km). At noon on the summer solstice, the sun was directly overhead in Syene, but in Alexandria it cast a shadow indicating an angle of about 7.2 degrees.

• Inputs: Distance = 800 km, Angle (θ) = 7.2°
• Calculation: Circumference = (800 km × 360) / 7.2° = 40,000 km
• Result: An Earth circumference of 40,000 km, remarkably close to the actual polar circumference of 40,008 km.

Example 2: A Modern Attempt

Imagine two schools on the same longitude, 500 miles apart. At local noon, one school measures a shadow from a 2-meter stick that is 0.352 meters long.

• Inputs: Distance = 500 mi, Stick Height = 2 m, Shadow Length = 0.352 m
• Angle Calculation: θ = arctan(0.352 / 2) = arctan(0.176) ≈ 10°
• Calculation: Circumference = (500 mi × 360) / 10° = 18,000 mi. This result is off, highlighting the importance of precision. Perhaps a visit to a right-triangle calculator could help refine the angle.

How to Use This Calculator for Calculating the Size of the Earth Using Trig

Follow these steps to perform your own calculation:

1. Enter Distance: Input the known north-south distance between your two measurement locations. The classic eratosthenes experiment used Alexandria and Syene.
2. Select Units: Choose whether your distance is in kilometers or miles. The output will match this unit.
3. Provide Object Height: Enter the height of your vertical object (gnomon).
4. Provide Shadow Length: Enter the length of the shadow cast by your object at local noon. Ensure the units for height and length are the same.
5. Analyze Results: The calculator instantly provides the calculated sun angle, and the resulting Earth circumference, radius, and diameter based on your inputs. The diagram will also update to reflect the angle.

Key Factors That Affect the Calculation

• Measurement Accuracy: Small errors in measuring the distance, stick height, or shadow length can lead to large inaccuracies in the final circumference.
• Locations on a Meridian: The calculation is most accurate when the two locations are perfectly North-South of each other (on the same longitude).
• Parallel Sun Rays: The method assumes the sun is so far away that its rays arrive at Earth in parallel. This is a very safe and accurate assumption.
• Simultaneous Measurement: Historically, the measurement had to be taken at the exact same local time (noon) on the same day. For Eratosthenes, this was when the sun was directly overhead in Syene.
• Earth’s Shape: The calculation assumes a perfect sphere. While the Earth is an oblate spheroid (slightly flattened at the poles), the spherical model provides a very close approximation for this method.
• Vertical Stick: The gnomon or stick must be perfectly perpendicular to the ground to get an accurate shadow measurement.

Frequently Asked Questions (FAQ)

Why do the object height and shadow length need to be in the same units?

Because they form two sides of a right triangle used to find an angle. The ratio must be dimensionless, so the units must cancel out. For unit conversions, you might use a distance converter.

What is the best day to perform this experiment?

The summer or winter solstice provides the most extreme sun angles, while the spring or autumn equinox can also be effective, especially if one location is on the equator.

How accurate is this method?

Amazingly accurate for its time. Eratosthenes’ calculation was within 1-2% of the correct value. Modern attempts can replicate this with careful measurement.

Can I do this with any two cities?

For best results, the cities should be on or very close to the same line of longitude, and you need to know the straight north-south distance between them.

What if my measurement is for the angle of the sun, not a shadow?

If you have the angle directly, you can mentally bypass the stick/shadow inputs. This calculator derives the angle from those inputs, as that is the classic method of trigonometry applications.

Does the curvature of the Earth affect the distance measurement?

Yes, the distance should be the “great-circle” distance along the curve of the Earth, not a straight line through it.

Why was Syene special?

Syene (modern Aswan) is located near the Tropic of Cancer. This meant that on the summer solstice, the sun was directly overhead at noon, casting no shadows. This created a zero-angle reference point for Eratosthenes.

What is a gnomon?

A gnomon is the part of a sundial that casts a shadow, but in this context, it refers to any vertical stick or pole used for the measurement.