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If you try to measure a circle with a rigid wooden ruler, you run into an immediate problem: straight lines cannot bend. Around 250 BCE in Syracuse, the Greek mathematician Archimedes found a brilliant way around this by asking how straight-edged polygons could trap a curve.
The mathematical constant Ο€\pi (pi) is defined as the ratio of a circle's circumference CC (the total distance around its perimeter) to its diameter d=2rd = 2r, expressed as Ο€=C2r\pi = \frac{C}{2r}.
Physical measurements with strings or wheels always have errors. How can we determine the value of Ο€\pi with absolute mathematical precision without measuring curves at all?

The Squeeze: Inscribed vs. Circumscribed

An inscribed polygon sits entirely inside a circle with all its corners touching the circle's boundary. A circumscribed polygon sits entirely outside, with each of its flat sides touching the circle at exactly one point (tangent to the circle).
Because the shortest path between any two points is a straight line, the perimeter of the inner polygon PinP_{\text{in}} must be strictly less than the circle's circumference CC, while the outer perimeter PoutP_{\text{out}} must be strictly greater.
Dividing every part of the inequality Pin<C<PoutP_{\text{in}} < C < P_{\text{out}} by the diameter 2r2r traps Ο€\pi between two exact limits: Pin2r<Ο€<Pout2r\frac{P_{\text{in}}}{2r} < \pi < \frac{P_{\text{out}}}{2r}
What do these perimeter bounds actually evaluate to when we build regular polygons step-by-step?

Worked Example: Hexagon Bounds (n=6n = 6)

Let's set our circle radius to r=1r = 1, making its diameter 2r=22r = 2. An inscribed regular hexagon divides into 6 equilateral triangles, each with side length equal to the radius (s=1s = 1), giving an inner perimeter of Pin=6Γ—1=6P_{\text{in}} = 6 \times 1 = 6.
For the circumscribed hexagon, the circle's radius r=1r = 1 is the perpendicular distance to each side (the apothem). Using a 30∘-60∘-90∘30^\circ\text{-}60^\circ\text{-}90^\circ triangle, half of each outer side is tan⁑(30∘)=13\tan(30^\circ) = \frac{1}{\sqrt{3}}, giving a full side length S=23S = \frac{2}{\sqrt{3}} and outer perimeter Pout=6Γ—23=43β‰ˆ6.928P_{\text{out}} = 6 \times \frac{2}{\sqrt{3}} = 4\sqrt{3} \approx 6.928.