Approximating π with Polygons

Discover how Archimedes used inscribed and circumscribed polygons to calculate the value of π over 2,000 years ago. Watch as increasing the number of sides brings us closer to the true value.

Polygon Visualization

Controls

Number of sides (n): 6

3 (Triangle)360 (Nearly Circle)

Circle diameter (d)

Show inscribed polygon

Show circumscribed polygon

Legend

Inscribed polygon

Circumscribed polygon

Circle

Calculated Metrics

Radius (r)

100.0

Diameter (d)

200.0

Side Lengths

s_inscribed = 2r × sin(180°/n)100.0000

s_{circumscribed} = 2r × tan(180°/n)115.4701

Perimeters

P_inscribed = n × s_in600.0000

P_{circumscribed} = n × s_out692.8203

π Approximations

π_inscribed = P_in/d3.000000

π_{circumscribed} = P_out/d3.464102

True π value3.141593

Error Analysis

Error (inscribed)0.141593

Error (circumscribed)0.322509

Bracket width0.464102

Current formulas (n = 6):

π_in = 6 × sin(180°/6) = 3.000000

π_out = 6 × tan(180°/6) = 3.464102

Conclusion

Archimedes' Bounds for π:

3.000000<π<3.464102

True π ≈ 3.141593

Bracket Width

0.46410162

Smaller is better

Quality Rating

Fair

Based on bracket width

Approximation Quality54%

🔍 Key Insight

Try increasing the number of sides to see better approximation!

📚 Remember

As n increases, the bounds tighten around π. This is because both polygons approach the shape of the circle, making their perimeters approach the circumference. The ratio of circumference to diameter is always π, regardless of the circle's size!

🏛️ The Mathematical Foundation

Archimedes' Method – Overview

Imagine a circle of radius r. Archimedes ingeniously:

Inscribed a regular polygon inside the circle (touches the circle at the vertices)
Circumscribed a regular polygon around the circle (each side is tangent to the circle)
Calculated the perimeters of both polygons
As the number of sides n increases, both perimeters converge to the circumference C = 2πr
Therefore: Perimeter_inscribed < C < Perimeter_{circumscribed}
Taking the ratio of each perimeter to the diameter gives lower and upper bounds for π

🎯 Key Insight

The ratio of circumference to diameter (C/d) is constant for all circles, and this constant is what we call π. By approximating the circumference with polygon perimeters, we can approximate π.

📐 The Mathematics

Inscribed Polygon

For a regular n-gon inscribed in a circle of radius r:

Side length: s_in = 2r × sin(180°/n)

Perimeter: P_in = n × s_in

π approximation: π_in = P_in/d

Circumscribed Polygon

For a regular n-gon circumscribed around a circle of radius r:

Side length: s_out = 2r × tan(180°/n)

Perimeter: P_out = n × s_out

π approximation: π_out = P_out/d

📊 Historical Example

Here's how the approximation improves as we increase the number of sides (for a circle with radius = 1):

Sides (n)	π_inscribed	π_{circumscribed}	Bounds on π
6	3.000	3.464	3.000 < π < 3.464
12	3.106	3.218	3.106 < π < 3.218
24	3.133	3.160	3.133 < π < 3.160
96	3.141	3.143	3.141 < π < 3.143

🏆 Archimedes' Achievement

Using 96-sided polygons, Archimedes determined that π is between 3.1408 and 3.1429 — accurate to two decimal places! This remained the most precise approximation of π for nearly 1,000 years.