How Fast Does the Gap Around Pi Shrink?

Polygon side counts doubling while the numerical gap between two estimates shrinks

We already know that more sides make a polygon follow a circle more closely.

Now let us ask a sharper question.

How fast does the error shrink?

Quick Answer

Double the number of sides. Once the polygons have enough sides, the gap between the two estimates becomes about one quarter as large.

sides × 2 → gap ≈ 1/4

The error does not just get smaller. It gets smaller in a pattern we can measure.

Which Gap Are We Measuring?

In the previous article, we measured a small radial gap between one polygon and the circle.

This article measures something different. We return to the two numerical fences around π.

Using modern trigonometry, a regular polygon with n sides gives:

lower = n sin(π / n)

upper = n tan(π / n)

Then:

gap = upper − lower

These formulas are a modern reconstruction of the polygon idea. Archimedes did not use modern sine, tangent, or this notation.

Look at the Pattern

Sides	Lower	Upper	Gap	New gap ÷ old gap
6	3.000000	3.464102	0.464102	—
12	3.105829	3.215390	0.109562	0.2361
24	3.132629	3.159660	0.027031	0.2467
48	3.139350	3.146086	0.006736	0.2492
96	3.141032	3.142715	0.001683	0.2498
192	3.141452	3.141873	0.000421	0.2499
384	3.141558	3.141663	0.000105	0.2500

Look at the last column.

The first ratio is about 0.236. Then it moves toward 0.25.

So the rule becomes clearer as the polygons get finer:

double the sides → about one quarter of the old gap

A ladder showing the numerical gap shrinking as polygon side counts double from 6 to 384

Figure 1. Each doubling makes the new gap approach one quarter of the previous gap.

Predict the Next One

At 384 sides, the gap is about:

0.000105

If the one-quarter pattern continues, what should happen at 768 sides?

0.000105 ÷ 4 ≈ 0.0000263

The actual modern calculation gives about:

0.00002628

The prediction is already very close.

Python — Modern Reconstruction

Python can repeat the calculation and compare each new gap with the one before it.

Python — watch the ratio move toward 0.25

import math

previous_gap = None

for n in [6, 12, 24, 48, 96, 192, 384, 768]:
    lower = n * math.sin(math.pi / n)
    upper = n * math.tan(math.pi / n)
    gap = upper - lower

    if previous_gap is None:
        print(n, gap)
    else:
        ratio = gap / previous_gap
        print(n, gap, ratio)

    previous_gap = gap

Predict first.

What number do you expect the ratio to approach?

Add 1536 to the list. Does the ratio move even closer to your prediction?

Why Does One Quarter Appear?

The table is enough to see the pattern. But modern calculus can describe it with one compact formula.

For large n, the gap behaves approximately like:

gap ≈ π³ / (2n²)

The important part is n².

If we double n, then n² becomes four times larger. A quantity divided by something four times larger becomes about one quarter as large.

n → 2n means n² → 4n²

That is why the one-quarter pattern appears.

The shape gets finer linearly, but this gap shrinks roughly with the square of the side count.

Why This Matters

“The answer gets better” is useful. But engineering and computation need a stronger question:

How quickly does it get better?

If we know how error changes when we refine a model, we can decide whether more work is worth it.

This way of thinking appears much later in numerical integration, grid refinement, and computer simulation. Those modern methods are not Archimedes' method. The connection is the habit of measuring how approximation error changes as a model is refined.

Where Archimedes Fits

Archimedes began with inscribed and circumscribed polygons and repeatedly doubled the number of sides, reaching 96 sides in his work on the circle.^[1]

His historical calculation did not use the modern sine and tangent formulas in this article. Those formulas let us reconstruct and extend the geometric pattern with today's notation.^[2]

Words to Keep

error
A measured difference between an approximation and the value or object we want.

ratio
One quantity divided by another.

refine
To make a model finer so it can represent something more closely.

rate
A way to describe how quickly something changes.

One Sentence to Keep

When the side count doubles, the gap between the polygon bounds becomes about one quarter as large.

What Should We Ask Next?

We now know two things.

The gap can become as small as we want. And we can measure how quickly it shrinks.

Long before modern calculus, mathematicians found a geometric way to use this idea.

Our next question is:

What Was the Method of Exhaustion?

Previous: Can a Polygon Ever Become a Circle?

Sources & Further Reading

François Dubeau, “Archimedes playing with a computer” — The Mathematical Gazette, 106(567), 2022.
“Archimedes' Recurrence Formula” — Wolfram MathWorld.
Archimedes, Measurement of a Circle, in The Works of Archimedes — edited and translated by Thomas L. Heath, Cambridge University Press.
“Regular Polygon” — Wolfram MathWorld; modern regular-polygon side and radius formulas.

Mathematical note: the trigonometric formulas, asymptotic formula, decimal table, and Python code are modern tools. They are used here to study the refinement pattern, not as Archimedes' original notation or calculation method.