A concise structural overview of a direct, physical method for determining the area of a circle without explicitly invoking the mathematical constant π. This repository contains a small static page and interactive materials exploring an alternative computational approach.
- Live demo: https://raphaelasanti.github.io/circle-area-without-pi/
- PDF: https://raphaelasanti.github.io/circle-area-without-pi/nopi.pdf
- Source & license: LICENSE
This project documents a physical approach and algebraic substitutions that produce the area of a circle without directly referencing π. Rather than relying on the well-known formula A = πr², this method uses measured geometric ratios and alternative constants to arrive at the same numeric result through a transparent, reproducible process. The material is grounded in practical measurement and suitable for educational exploration, computational verification, and deepening understanding of how circle geometry relates to numeric constants.
- Reframe circle-area computation using measured ratios and alternative constants derived from physical geometry.
- Substitute π systematically so the final numeric evaluation does not require writing the symbol explicitly.
- Verify using reproducible, physical measurement and a numeric recipe suitable for practical computation.
- Understand the underlying relationships between circle dimensions without abstracting away the concrete steps.
- Students and educators exploring geometric reasoning beyond standard formulas.
- Mathematicians and researchers interested in alternative framings of classical geometry.
- Practitioners seeking transparent, auditable methods for numeric computation.
- Curious minds investigating the role of constants in mathematical description.
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Visit the online page: https://raphaelasanti.github.io/circle-area-without-pi/
— Interactive walkthrough of the method with step-by-step substitutions. -
Download/read the PDF: https://raphaelasanti.github.io/circle-area-without-pi/nopi.pdf
— Complete reference with worked numeric examples and detailed derivations. -
Try it yourself: Use the provided tools to compute circle area for a measured radius or circumference, then verify against conventional formulas.
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Ask questions: For discussion or clarification, open an issue in this repository.
Example workflow: Measure the radius of a circle → Apply the substitution steps shown in the PDF → Calculate using only basic arithmetic → Compare to πr² for verification.
Improvements are welcome. Suggested ways to contribute:
- Fix typos or clarify explanations (open a PR).
- Add explicit numeric examples or interactive demos in JavaScript/HTML.
- Report inaccuracies or suggest research citations (open an issue first).
- Add pedagogical materials (lesson plans, classroom examples).
Before proposing any changes, please open an issue describing your suggestion.
This work is sustained by voluntary contributions.
- Cash App: https://cash.app/$phirunltd
- Gumroad (optional donation): https://phirun4.gumroad.com/l/zsmzjh
- Free alternative: The same eBook is available in the the-end-of-time repository.
Raphael E. Asanti — July 2026
License: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0
See LICENSE for full details.