Look at a hexagon. No one would mistake it for a circle.
Now imagine a polygon with 96 sides. From a distance, its edge can look almost round.
So what changed? And if we keep adding sides, what exactly gets closer?
Quick Answer
Each new side makes the straight pieces shorter. Shorter pieces can follow the curve more closely.
For a polygon inside a circle, the perimeter grows toward the circle's circumference. For a polygon outside the circle, the perimeter shrinks toward it.
More sides do not make the curve disappear. They make the straight-line approximation fit the curve more closely.
Start with Six Sides
Imagine a regular hexagon drawn inside a circle. Its six corners touch the circle.
But each side is straight. The circle between two corners bends outward.
So each straight side takes a shortcut across the curved arc. Add the six shortcuts together, and the hexagon's perimeter is too short.
Think before you read on.
If we split every side into smaller pieces, should those new straight pieces follow the circle better or worse?
Now Double the Number of Sides
Go from 6 sides to 12. Then from 12 to 24. Then to 48 and 96.
6 → 12 → 24 → 48 → 96 → 192
The important change is not simply that there are more lines. Each line also covers a smaller part of the circle.
A short straight segment can match a small piece of a curve better than a long one can. That is why the polygon begins to look smoother.
More sides → shorter straight pieces → smaller mismatch.
Two Polygons Give Us Two Fences
In the previous article, we used two polygons. One sat inside the circle and one sat outside it.
The inside polygon gave a value that was too small. The outside polygon gave a value that was too large.
That means the true circle value stayed between them.
lower estimate < π < upper estimate
As the number of sides increases, the lower estimate rises and the upper estimate falls. The two fences move closer together.
Watch the Gap Shrink
We can measure the space between the two fences. Call it the gap.
gap = upper estimate − lower estimate
The table below uses modern trigonometry to reconstruct the polygon idea. The values are normalized so they can be compared directly with π.
| Sides | Lower | Upper | Gap |
|---|---|---|---|
| 6 | 3.000000 | 3.464102 | 0.464102 |
| 12 | 3.105829 | 3.215390 | 0.109562 |
| 24 | 3.132629 | 3.159660 | 0.027031 |
| 48 | 3.139350 | 3.146086 | 0.006736 |
| 96 | 3.141032 | 3.142715 | 0.001683 |
| 192 | 3.141452 | 3.141873 | 0.000421 |
Look down the last column. The gap becomes much smaller as the side count grows.
At 6 sides, the two fences are far apart. At 192 sides, both values already begin with 3.141.
This Is an Approximation
An approximation is a value or model that is not exact, but is close enough to help us understand or calculate something.
The polygon does not have a curved edge. It still has straight sides.
But by making those sides shorter and more numerous, we can make the polygon follow the circle more closely.
This gives us a powerful way to work with something difficult: replace it with something simpler, then improve the replacement.
A Modern Way to Test the Pattern
We can make the computer repeat the same modern reconstruction for several polygons. This time, instead of changing one value and running the program again, we will let Python repeat the calculation for us.
import math
for n in [6, 12, 24, 48, 96, 192]:
lower = n * math.sin(math.pi / n)
upper = n * math.tan(math.pi / n)
gap = upper - lower
print(n, lower, upper, gap)
The for line simply repeats the same calculation
for each number of sides in the list.
Try this.
Before you run the code, predict what the last column will do. Will the gap grow or shrink?
Then add 384 to the list.
Does the pattern continue?
A Historical Note
The sine and tangent formulas in the code are a modern reconstruction. Archimedes did not use modern trigonometric notation.
In Measurement of a Circle, his reasoning was geometric, and his famous calculation reached polygons with 96 sides.[1]
Modern mathematical treatments show rigorously how the perimeters of regular inscribed and circumscribed polygons approach the circumference as the polygons are refined.[2]
Why This Idea Matters
A circle is smooth. A polygon is made from straight pieces.
Yet the simpler shape can still help us understand the harder one. We only need a way to make the approximation better and to measure whether it is improving.
That habit appears again and again in mathematics and computation: build a simpler model, refine it, and compare the result.
The modern methods are not the same as Archimedes' geometry. The connection here is a problem-solving idea, not a claim of direct historical identity.
Words to Keep
approximation
A value or model that is close to something we want to understand.
perimeter
The total distance around a polygon.
gap
The difference between the upper and lower estimates in this experiment.
refine
To make a model finer or more detailed so it can represent something more closely.
One Sentence to Keep
As a regular polygon gets more sides, its shorter straight pieces follow the circle more closely, and the gap between the two estimates shrinks.
What Should We Ask Next?
A 192-sided polygon can look almost like a circle. We could keep going to 384 sides, 768 sides, or far beyond.
But there is a deeper question hiding here:
Can a polygon ever actually become a circle?
That question will force us to separate two ideas that look similar at first: getting arbitrarily close and becoming exactly the same thing.
Previous: How Did Archimedes Calculate Pi Without a Calculator?
Sources & Further Reading
- Archimedes, Measurement of a Circle, in The Works of Archimedes — edited and translated by Thomas L. Heath, Cambridge University Press.
- David Weisbart, “Modernizing Archimedes’ Construction of π” — Mathematics, 8(12), 2204, 2020.
- “A History of Pi” — MacTutor History of Mathematics, University of St Andrews. The modern trigonometric formulas are used there as a reconstruction of Archimedes' geometric process.
Historical note: the decimal values, trigonometric formulas, and Python code in this article are modern tools used to explore the geometric idea. They are not presented as Archimedes' original notation or calculation method.