The radius (r) of a circle is the distance set on your compass when you draw a circle, and the distance from the center to the boundary of any circular object. The distance around, or perimeter, is called the circumference (C), and the straight-line distance across the circle, through the center, is called the diameter. The diameter (d) is exactly twice the radius. There is a fixed ratio between the circumference (C) and the diameter (d), expressed algebraically as ^C⁄_d. What, exactly, is the value of this ratio?

Fig. 5-1

The number must be less than 4, because the perimeter of a square that encloses a circle is 4d. See Figure 5-1. The number must be greater than 3, because when 6 radii are marked off on the circumference, the straight-line figure inside is a regular hexagon, with perimeter 3d.

Archimedes of Syracuse (Sicily, not New York) was curious about the precise ratio between C and d. Around 250 BC, he had no calculator and not even a good number system, but by constructing polygons with more-and-more sides, and dividing them into triangles, he found a good approximation of the number, which about 1700 AD came to be called pi (π) the 16th letter of the Greek alphabet.

^C⁄_d = π, C = πd = 2πr

3¹⁄₇ = ²²⁄₇ > π > 3 ¹⁰⁄₇₁ = ²²³⁄₇₁

The ratio of circumference to diameter of a circle (π) is given here as approximately 3¹⁄₇, which is the same as ²²⁄₇. It is possible to have fractions between any two whole numbers, not just between 0 and 1. In principle, the interval between 3 and 4 could be divided in 71 equal pieces, and we could take 10 of them to make the desired ratio. Thus π is also about 3¹⁰⁄₇₁.

Although fractions can be precise, and operations with fractions have simple, well-defined rules, it is often more practical to do calculations with their decimal equivalents: ¹⁄₄ = 0.25, ¹⁄₈ = 0.125, ¹⁄₁₆ = 0.0625 etc. The decimal equivalent of a fraction is the upper number (numerator) divided by the lower one (denominator).

An average of these two values is very close to the desired ratio. A closer ratio, ³⁵⁵⁄₁₁₃, was known to ancient Chinese astronomers. The value in a scientific calculator is 3.141592654. By playing with numbers, one can find fractions even closer, such as ^{12 546 274}⁄_{3 993 603.}

The value of π can be calculated ever more closely by adding various series of ever-smaller fractions (discovered by Leibniz and Euler), a tedious procedure ideally suited to computers. Some people have become so obsessed with π that they carry calculations to ridiculous extremes. That is why π is called an irrational number. No, not really; it is because there is no ratio of whole numbers that expresses it exactly.

Fig. 5-3

photo credit: David Spender

The area of a circular space, such as a ring at a three-ring circus, or the surface of Crater Lake, should be some multiple of its linear dimension squared. Perhaps the area of a circle is some multiple of r². Eureka!

A = πr²

The childish geometrical joke is: “Pie r not square, pie are round.” In most languages, however, i has the sound of i in machine. With an international group, the joker should say, “pea are round.”

Hagia Sophia in Istanbul (Fig. 5-3) was first built as a church by Constantius in 360AD, was burned and rebuilt twice, became a mosque, and is now a museum.

A sphere is the solid figure generated by rotating a circle, and there are many interesting sphere-like objects, such as Earth, for example. Geometry of the sphere was investigated by Archimedes, and of course the constant π appears in the equations for volume and surface of the sphere. See C. 8.