Pi Explained: Circumference, Area and Archimedes

The same number in every circle

Measure around the edge of a round table, then across it through the middle. Divide the first number by the second. Do the same with a coin, a plate and a tyre, and you always get about 3.14. The edge is the circumference C, the distance across is the diameter d, and the ratio C / d is the same for every circle there is. We call it π, the Greek letter pi, and π ≈ 3.14159. So C = π × d. A table 1.2 m across has an edge of 1.2 × 3.14159 ≈ 3.77 m. The letter was first used for this number by William Jones in 1706 and was made famous by Euler.

A wheel that rolls

Imagine a wheel with a diameter of exactly 1 and a dot painted on its rim. Roll it along a straight line for one full turn. The wheel travels a distance equal to its circumference, which is π × 1, about 3.14159. In the drawing the line unrolls beside the wheel: three whole widths of the wheel and a little more, 0.14159 to be exact. That leftover never fits a whole number of times. A bicycle wheel works the same way. With a diameter of 0.7 m, ten turns carry you 10 × 0.7 × 3.14159 ≈ 22 m.

Archimedes traps pi

Nobody can measure π exactly with a tape. Around 250 BC Archimedes of Syracuse found a clever way to trap it. Draw a polygon just inside the circle and another just outside. The circle's edge is longer than the inner polygon's perimeter and shorter than the outer one's. With a hexagon the bounds are crude. But keep doubling the sides, 6, 12, 24, 48 and then 96, and the two polygons squeeze closer. For a circle of diameter 1, the 96-sided polygons have perimeters of about 3.1410 inside and 3.1427 outside.

The famous bounds

From his 96-sided polygons Archimedes proved that 3 10/71 < π < 3 1/7. In decimals that is 3.14085 < π < 3.14286, and the true value, 3.14159, sits comfortably inside. The well-known fraction 22/7 is just the upper bound, and it is right to two decimal places. Archimedes had no decimals: he worked with fractions and with square roots that he bounded by hand. In the sandbox slide the number of sides from 3 to 96 and watch the inner and outer bounds close in on the circle. A better fraction, 355/113 ≈ 3.1415929, was found by the Chinese mathematician Zu Chongzhi in the fifth century.

The area of a circle

How big is the inside of a circle? Cut it like a pizza into many thin wedges, then lay them side by side, one pointing up and the next pointing down. The result looks like a bumpy rectangle, and the more wedges, the straighter it gets. Its long side is half the circumference, π × r, and its height is the radius r. So the area is π × r × r, written A = π r². For a pizza with radius 15 cm, A = π × 225 ≈ 707 cm². A surprise: one pizza 30 cm across gives about 707 cm², while two pizzas 20 cm across give only 2 × 314 = 628 cm². Try other radii in the sandbox.

A number without end

Is π a fraction? No. In 1761 Johann Lambert proved that π is irrational: no fraction, however big its numbers, equals it. So its decimals never end and never settle into a repeating pattern. In 1882 Ferdinand von Lindemann went further and showed that π is transcendental, which also proved that an ancient puzzle, drawing with ruler and compass a square of the same area as a given circle, is impossible. The digits begin 3.14159265358979323846. For real work you need few of them: NASA's Jet Propulsion Laboratory uses fifteen decimals, 3.141592653589793, to steer spacecraft around the Solar System.

Digits by the trillion

Why do people keep computing π, then? Partly to test computers and programs, and partly for the love of it. In 1706 John Machin reached 100 digits by hand, using a clever formula. The ENIAC computer produced 2 037 digits in 1949, and in 2022 a team using Google Cloud computed 100 trillion. Yet no amount of digits changes the first idea of this chapter: for every circle, C / d is π. And because the digits never repeat, you can never write π down in full.

Pi and the circle