The Number π and Its Link to the Circle

Measure any circle in the universe, divide its circumference by its diameter, and you always get the same number. That constant is π — and mathematicians have chased its digits for more than two thousand years.

Concept The ancient observation

Lay the diameter along the circumference and it fits three times, with a little left over. That leftover is about 0.14159 of a diameter — and it never changes, no matter how large or small the circle.

Theorem The defining ratio

\pi = \frac{C}{d}

Rearranged, this gives the familiar circumference formula $C = \pi d$ . Since the diameter is twice the radius, it is also written $C = 2\pi r$ .

Concept Why π is irrational

π is an irrational number: its decimal expansion runs forever without ever settling into a repeating pattern.

\pi = 3.14159265358979\ldots

Compare

\tfrac{1}{3} = 0.333\ldots

, where the 3 repeats forever.

π has no such repeating block.

This is why no fraction can capture it exactly. The familiar $\tfrac{22}{7} \approx 3.142857$ is a useful approximation, not the true value.

Example Finding a circumference

A circular table has a diameter of 2 metres. How far is it around the edge?

C = \pi d

C = \pi \times 2

C \approx 3.14159 \times 2 = 6.28318

⟹ about 6.28 metres

Note The long hunt for digits

Around 250 BC, Archimedes trapped π between polygons drawn inside and outside a circle, reaching roughly 3.1416.

About 480 AD, Zu Chongzhi in China obtained seven correct decimal places.

In the 1700s and 1800s, infinite series pushed the count into the hundreds, all by hand.

Modern computers have now calculated tens of trillions of digits.

The irony is that precision on this scale is never needed in practice — space agencies work comfortably with about 15 decimal places. The pursuit has always been about testing methods and machines, not about the circle.

Summary

π is the ratio of circumference to diameter, identical for every circle.
π = C / d, which rearranges to C = πd = 2πr.
Its value is about 3.14159, and 22/7 is only an approximation.
π is irrational: the decimals never end and never repeat.
Computing its digits drove mathematics from Archimedes to the supercomputer.