Squaring the Circle Revisited

TL;DR

The paper addresses the classical problem of squaring the circle by focusing on high-precision constructible approximations to $\pi$. It presents two explicit straightedge-and-compass constructions: a new geometric route to the length $\sqrt{\frac{6}{5}(1+\phi)}$ and a nine-decimal-accurate construction for $\sqrt{\frac{63}{25}\left(1+\frac{5}{2}\cdot \frac{15\sqrt{5}-7}{269}\right)}$, thereby linking circle area to algebraic expressions involving the golden ratio $\phi$ and $\sqrt{5}$. The first construction yields a squared-length approximation to $\pi$ with ratio $\frac{6}{5}(1+\phi)/\pi\approx 1.000015$, while the second achieves an exceptionally tight error bound of $1.000000000068$ (with $10^{12}$ parts considered). Collectively, these results extend Ramanujan’s and Dixon’s historical approximations with transparent geometric procedures and explicit error quantification, contributing practical, constructible means to approximate circle area with high precision.

Abstract

Squaring the circle is impossible, but it can be squared approximately. Ramanujan gave a construction correct to eight decimal places. In his book Mathographics, Dixon gave constructions correct to three decimal places. In this article, we provide a new construction correct to three decimal places and another correct to nine decimal places.

Figures (2)

• Figure 1: The shaded square and the shaded circle almost have the same area.
• Figure 2: Construction of $\frac{\sqrt{63}}{5}$, $\frac{\sqrt{15\sqrt{5}-7}}{5}$, and $\frac{\sqrt{269}}{8}$

Theorems & Definitions (4)

• proof
• Remark 2.1
• proof
• Remark 3.1