Approximating π by Slicing the Circle - from Liu Hui, an acient Chinese mathmaticians

May 11, 2025

In the 3rd century, Chinese mathematician Liu Hui Wikipeida approximated π, by inscribing regular polygons in a circle and letting the number of sides double.

Let’s recreate his method in Python!

Play with the code, by changing the “n” variables in the code below, either at my google colab page.

Leave your comments below!

Python Code

# Demo of Liu Hui’s Inscribed Polygon π-Approximation
# Description: use 6,12,24... polygons to inscribe circle, for pi caluation
# Author: Daniel ge
# Date:   2025-05-11
import numpy as np
import matplotlib.pyplot as plt

def draw_inscribed(n):
    angles = np.linspace(0, 2*np.pi, n+1)
    x, y = np.cos(angles), np.sin(angles)
    plt.plot(x, y, '-o', ms=3, label=f"{str(int(n))}-gon")
    # outline to inscribe circle
    theta = np.linspace(0, 2*np.pi, 300)
    plt.plot(np.cos(theta), np.sin(theta), color='gray')

plt.figure(figsize=(9,6))
for n in [6, 12, 24, 36, 48]:  # 6, 12,24,36,48 side polygons
    draw_inscribed(n)
plt.axis('equal'); plt.axis('off')
plt.title("Inscribed Polygons Approaching the Circle")
plt.legend(loc='lower right')
plt.show()

# Approximation accuracy
ns = [6 * 2**k for k in range(7)]  # [6,12,24,...,1536]
perimeters = [n * 2 * np.sin(np.pi / n) for n in ns]
approximations = [P/2 for P in perimeters]

plt.plot(ns, approximations, 'o-', label="P_n/2")
plt.axhline(np.pi, color='gray', linestyle='--', label="π")
plt.xscale('log', base=2)
plt.xlabel("Number of sides (n)")
plt.ylabel("Approximation of π")
plt.title("Liu Hui’s Inscribed Polygon π-Approximation")
plt.legend()
plt.grid(True, ls='--', alpha=0.5)
plt.show()