Walking to the Horizon To Find the Earth’s Radius

Introduction

Chamberland [2025a] offered an ingenious and innovative technique for calculating the radius of the Earth, based on walking across a flat surface (in his case, the Bonneville Salt Flats in Utah). But his estimate wound up being 8% too small. We investigate why.

Note: The information below was created with the assistance of AI.

Level of Mathematics
Overall Level:

Advanced High School → Early Undergraduate

Evidence:

• Uses:
• Geometry of circles and triangles
• Algebraic manipulation
• Approximations (Taylor series)
• Introduces:
• Error analysis
• Sensitivity of models
• Basic physics integration

Mathematical sophistication:

• Trigonometric approximations:
• 
• Derived formula:
• 
• Algebraic modeling with assumptions and approximations

Interpretation:

• Accessible to:
• Precalculus / AP Calculus students
• Intro college quantitative reasoning or modeling courses

Subject Matter
Core Mathematical Topics:

• Geometry
• Circles, arcs, tangents (see Figures 1–3, pages 2–4)
• Trigonometry
• Cosine relationships and small-angle approximations
• Algebraic Modeling
• Deriving formulas from physical situations
• Approximation Methods
• Taylor series approximations
• Error & Sensitivity Analysis
• Propagation of measurement error

Supporting Topics:

• Scientific modeling
• Dimensional reasoning
• Basic physics (light, refraction)

Application Areas
Primary Application:

• Geophysics / Earth measurement
• Estimating Earth’s radius using observation

Secondary Applications:

• Physics
• Optics (refraction, index of refraction)
• Experimental design
• Measurement techniques and uncertainty
• Navigation / surveying

Real-world relevance:

• Demonstrates how simple measurements → global-scale estimates
• Connects math to:
• Astronomy
• Earth science
• Engineering measurement

Prerequisites
Required Background:

Mathematics:

• Algebra:
• Manipulating equations
• Geometry:
• Circles, triangles
• Trigonometry:
• Cosine function

Recommended:

• Basic calculus concepts:
• Idea of Taylor approximation
• Understanding of:
• Rates of change (informal)
• Approximations and limits

Not required:

• Advanced calculus
• Differential equations

Correlation to Mathematics Standards
US Common Core (High School)

Strong alignment with:

HSG-C (Geometry: Circles)

• Arc length and radius relationships

HSG-SRT (Similarity & Trigonometry)

• Trigonometric ratios and relationships

HSA-CED (Create equations)

• Modeling real-world phenomena

HSM (Modeling Standard)

• Full modeling cycle:
• Assumptions → approximation → refinement

AP Courses
AP Precalculus / AP Calculus AB

• Trigonometric approximations
• Modeling with functions
• Intro to Taylor approximations

AP Physics

• Refraction and optics (qualitative)

Undergraduate Standards
Aligned with:

• Quantitative reasoning courses
• Mathematical modeling (intro level)
• Applied mathematics / physics courses

Mathematical Practices (Process Standards)
This module strongly emphasizes:

• MP4: Model with mathematics
• Core focus: real experiment → mathematical model
• MP2: Quantitative reasoning
• Interpreting physical meaning of variables
• MP3: Critique reasoning
• Explaining why the model was wrong (8% error)
• MP6: Precision