Circles, radius, and pi

In brief

Wrap a string once around a bicycle wheel, then straighten the string out and measure it against the wheel's diameter, straight across through the center. Do this for a bicycle wheel, a dinner plate, and a coin, and the same ratio turns up every time: the string is a little more than three times the diameter, always the same little more, no matter how big or small the circle. That fixed ratio is pi, and the fact that it is fixed at all, that size drops out completely, is the entire content of this entry: pi is not a decimal to memorize but the signature of every circle's own proportionality.

The full treatment

First look: measuring circles of different sizes

Take three circular objects of obviously different sizes, a coin, a dinner plate, and a car tire, and for each one measure its circumference, the distance around, and its diameter, the distance straight across through the center. Divide circumference by diameter for each object. Every division gives close to 3.14, whether the object is a few centimeters or nearly a meter across. Nothing else in ordinary geometry behaves like this in a surprising way: double the side of a square and its perimeter doubles as expected, and the ratio of perimeter to side stays a fixed 4 regardless of size too, so a fixed ratio alone is not remarkable. What is remarkable about the circle is that this fixed ratio, pi, is not four, or three, or any tidy fraction: it is a specific, irrational, never repeating number that nobody chose, and every circle, without exception, obeys it.

Building the idea: radius as the one number that defines a circle

A circle is defined as the set of all points at a fixed distance from a single center point. That fixed distance is the radius. Nothing else about a circle's shape is free to vary once the radius is fixed: two circles with the same radius are, in every geometric respect, identical, just possibly moved to different places. This is worth pausing on, because it means every property of a circle, its circumference, its enclosed area, must be expressible purely in terms of its one defining number, the radius, or equivalently its diameter, twice the radius.

The formal relationship: defining pi and deriving area

Define pi precisely as the ratio of any circle's circumference to its diameter: pi equals circumference divided by diameter. This definition already forces the circumference formula: circumference equals pi times diameter, or, using radius, circumference equals two times pi times radius, since diameter is twice the radius.

The area formula takes more building. Here is a way to see why area equals pi times radius squared, using only the circumference relationship just derived and a dissection. Slice a circle of radius r into a large number of thin, equal, pie shaped wedges, like a pizza cut into many slices, then lay the wedges down in a row, alternating point up and point down, so they interlock into a shape close to a parallelogram, or more precisely, close to a rectangle as the number of slices grows large. The row's length is half the total circumference, because each wedge contributed its curved edge to alternating sides, and half of the total curved boundary lands along the top of this near rectangle while the other half lands along the bottom. The row's height is very close to the radius r, the distance from the circle's center out to its rim, since that is the straight length of each wedge's two straight edges. So the near rectangle has width one half of the circumference and height r; its area is one half of the circumference, times r. Substitute the circumference relationship, circumference equals two pi r: one half of two pi r, times r, equals pi r times r, equals pi r squared. As the wedges are cut thinner and more numerous, the pieced together shape's wiggly top and bottom edges straighten into genuine straight lines, and the approximation becomes exact in the limit, giving the area of a circle of radius r as exactly pi times r squared.

What pi is and is not

Pi is a single fixed number, approximately 3.14159, that emerges purely from the geometry of flatness: it is provably irrational, meaning it cannot be written as one whole number divided by another, and it is provably transcendental, meaning it is not the root of any polynomial equation with whole number coefficients, a fact proved by Ferdinand von Lindemann in 1882. Its value cannot be derived from any simpler number; it must be computed, and it has been computed to trillions of digits.

Lineage

Approximations to pi appear across independent ancient traditions: the Rhind Mathematical Papyrus from Egypt, around 1650 BCE, implies a value close to 3.16, and Babylonian tablets used 3 and one eighth. Archimedes of Syracuse, in the third century BCE, gave the first rigorous method for bounding pi, by inscribing and circumscribing regular polygons of increasing numbers of sides around a circle and computing their perimeters, proving pi lies between 3 and ten seventy firsts and 3 and one seventh. This method, refined over centuries, is the ancestor of the wedge dissection given above. Independently, the Chinese mathematician Liu Hui in the third century CE, and later Zu Chongzhi in the fifth century, extended polygon methods to remarkable precision. The Greek letter pi itself was popularized as notation by William Jones in 1706 and cemented by Leonhard Euler later in the eighteenth century.

The strongest case for it

The ratio's fixed value is not an empirical regularity that might one day break; it is a consequence of what a circle is, the set of points a constant distance from a center, together with the flatness of the plane it sits in, and it has been proved, not merely observed, since Archimedes. Every measured circle in engineering, astronomy, and manufacturing confirms it to the precision of the measuring instrument, with zero known exceptions in flat space. The relationship's reach is enormous precisely because circles and rotations are everywhere: wheels, orbits, oscillations, and waves all inherit pi through the same geometry, and the same constant that relates a coin's rim to its width also governs the period of a pendulum and the phase of an alternating current.

The strongest case against it

Pi is a fact about flat, Euclidean circles specifically; draw a circle on the surface of a sphere, all points a fixed distance from a center measured along the surface, and the ratio of its circumference to its diameter is smaller than pi, and depends on how big the circle is relative to the sphere, because the surface is curved. A common misconception is treating pi as merely a rounding convenience, 3.14, rather than an exact, irrational value; using a truncated decimal introduces a real, compounding error in any calculation involving very large radii, such as astronomical distances. Another misconception is assuming the circumference to diameter ratio and the area to radius squared ratio are independent facts to memorize separately, when one is derivable from the other exactly as shown above; treating them as two unconnected formulas hides the reason area involves pi at all.

Where it stands now

The value, irrationality, and transcendence of pi are fully settled mathematics, proved rigorously and not subject to revision. What remains active is purely computational and cultural: the ongoing effort to compute more digits of pi, now in the tens of trillions, which tests supercomputing methods and number theoretic algorithms but has no bearing on any application, since a few dozen digits already exceed the precision of any physical measurement ever made.

Test yourself

You are given a circular water tank of unknown size, standing on a factory floor, and you may only wrap a tape measure around its outside and measure straight across a visible opening at the top. Using the relationships developed here, describe how to compute the tank's radius, its cross sectional area, and the volume of water it holds per unit of height, and state explicitly which measurement, circumference or diameter, you trust less if the tank's rim is slightly dented, and why that dent affects the two measurements differently.