Measuring Earth's Radius and Circumference Using a Gnomon
This paper documents a hands-on experiment to estimate Earth's radius and circumference using the ancient gnomon shadow method, replicating the approach first used by Eratosthenes. By measuring the length of a gnomon's shadow at local solar noon on September 17th in New Mexico and comparing it to a known solar angle, the student calculates an approximate Earth radius of 6,610 km — within a small percentage of the accepted equatorial radius of 6,378 km. The paper also addresses sources of measurement error, the importance of taking readings at local solar noon, and the potential application of this technique on other planetary bodies such as the Moon and Mars.
- Overview of the Measurement Method: Gnomon shadow taken at solar noon in New Mexico
- Calculating the Angle and Earth's Radius: Trigonometric steps to derive Earth's radius
- Comparing Results to Accepted Values: Calculated vs. accepted equatorial and polar radii
- Sources of Error in Shadow Measurements: Why solar noon matters and timing error effects
- Applying the Technique to Other Planetary Bodies: Using gnomon method on Moon and Mars
✍️ How to write this paper — guide, tools & examples ▾
What makes this paper effective
- It clearly documents each step of the experimental calculation, making the methodology transparent and reproducible.
- The paper demonstrates honest error analysis by comparing the calculated result to both polar and equatorial reference values and explaining which is more appropriate for the geographic context.
- The Q&A format used for the error discussion effectively isolates specific sources of inaccuracy, helping readers understand how timing affects results.
Key academic technique demonstrated
The paper demonstrates applied trigonometry in a real-world scientific context. Rather than simply reporting a formula, the student shows each computational step — from the arcos calculation through the radius formula — and connects it to a physical observation. This step-by-step transparency is a hallmark of strong lab-report writing at the undergraduate level.
Structure breakdown
The paper opens with the experimental setup and raw data, moves through the calculation sequence, then benchmarks the result against reference values. It closes with a conceptual Q&A section that probes the method's limitations and broader applicability. This progression from data to analysis to reflection follows a standard scientific report structure.
Overview of the Measurement Method
To estimate the radius and circumference of Earth, this experiment used a gnomon to measure the length of the Sun's shadow on September 17th. This approach replicates the classic method attributed to Eratosthenes, who used shadow angles at two locations to infer Earth's curvature. In this experiment, the gnomon had a height of 50 cm, and the measured shadow length at local solar noon was approximately 39.8 cm.
Calculating the Angle and Earth's Radius
The Sun angle (θ) at the measurement site was found using the inverse cosine (arcos) of the ratio of the gnomon height to the hypotenuse formed by the gnomon and its shadow:
θ = arcos(30.3 / 50)
This yields an angle of approximately 4.49 degrees of arc difference between the two locations. Since the known solar angle for September 17th at the New Mexico site is 57.19 degrees, this value was used to calculate Earth's radius with the following formula:
R = (L × 360) / (θ × 2π)
where L = 518 kilometers (the distance between Denver, CO and the New Mexico measurement site), and θ = 4.49 degrees:
R = (518 × 360) / (4.49 × 3.14159 × 2)
R ≈ 6,610.075 kilometers
The circumference was then calculated using the standard formula C = 2πR:
C = 2 × 3.14159 × 6,610.075 ≈ 41,532.291 kilometers
Comparing Results to Accepted Values
Reference values for Earth's radius are:
Polar radius: R = 6,356.755 km
Equatorial radius: R = 6,378.140 km
Because both measurement locations — Denver, CO and the New Mexico site — are geographically closer to the equator than to the North Pole, the equatorial radius is the appropriate benchmark for comparison. The percentage error is calculated as follows:
Error = |6,378.140 − 6,610.075| / 6,378.140 ≈ 3.6%
This relatively small error demonstrates the viability of the gnomon method for estimating Earth's dimensions with simple equipment.
Always verify citation format against your institution’s current style guide requirements.