That every circle has the same ratio between circumference and diameter was known before Archimedes. What was uncertain was the value of this ratio, which we now call $\pi$: the Babylonians sometimes used $3\frac{1}{8}$, the Egyptians something close to $3.16$. In the short treatise Measurement of a Circle, Archimedes did two things: he related the area of the circle to its circumference, and he showed how to compute $\pi$ to any desired precision.

The area of the circle

The first proposition says that the area of a circle equals that of a right triangle whose legs are the radius $r$ and the circumference $C$:

$$A = \frac{1}{2}\,r\,C.$$

The idea is to cut the circle into very thin slices, like a pizza, and notice that each slice is almost a triangle of height $r$. Archimedes gives a rigorous proof by exhaustion: he shows that the area can be neither larger nor smaller than that of the triangle. Since $C = 2\pi r$, the formula becomes the familiar $A = \pi r^2$.

Trapped between two polygons

To estimate $\pi$, Archimedes inscribed and circumscribed regular polygons in the circle. The perimeter of the inner polygon is less than the circumference; that of the outer polygon is greater. Dividing by the diameter, $\pi$ is trapped between two numbers that can be computed.

He began with hexagons. The inscribed hexagon has perimeter $6r$, that is, $3$ diameters, and the circumscribed one has perimeter $4\sqrt{3}\,r \approx 6.93\,r$. Then he doubled the number of sides: 12, 24, 48 and finally 96. At each step he used a geometric relation between the polygon with $n$ sides and the one with $2n$ sides.

The final result was

$$3 + \frac{10}{71} < \pi < 3 + \frac{1}{7}, \qquad \text{that is,} \qquad 3.1408 < \pi < 3.1429.$$

The upper bound, $\frac{22}{7}$, became so famous that it is still used today as the school approximation of $\pi$.

Calculations without a calculator

The most impressive part is arithmetical. Archimedes had no decimal numbers and no practical notation for fractions, and yet he needed square roots with many digits. Right at the start he used

$$\frac{265}{153} < \sqrt{3} < \frac{1351}{780},$$

excellent approximations ($1.73203 < \sqrt{3} < 1.73205$), without explaining how he got them. Over the four doublings he always chose to round on the safe side: down for the inner polygon, up for the outer one, so that the inequalities stayed true.

Exercise: the hexagon and the square

Show that the regular hexagon inscribed in a circle of radius $r$ has perimeter $6r$ and that the circumscribed square has perimeter $8r$. What bounds for $\pi$ do they give?

The inscribed hexagon divides into 6 equilateral triangles of side $r$, so each side is $r$ and the perimeter is $6r$. The circumscribed square has side equal to the diameter, $2r$, and perimeter $8r$. Since $6r < 2\pi r < 8r$, we get $3 < \pi < 4$. It is the beginning of the method; polygons with more sides narrow the interval.

Exercise: how good is the average?

Compute the average of Archimedes’s two bounds, $3\frac{10}{71}$ and $3\frac{1}{7}$, and compare it with $\pi \approx 3.14159$.

$3\frac{10}{71} \approx 3.140845$ and $3\frac{1}{7} \approx 3.142857$. The average is $\approx 3.141851$, which differs from $\pi$ by about $0.0003$. The relative error is one part in ten thousand: enough for any construction in antiquity.

Sources: Archimedes, Measurement of a Circle, propositions 1–3 · Thomas L. Heath, The Works of Archimedes (1897) · Petr Beckmann, A History of π (1971) · Jonathan M. Borwein, “The Life of Pi: From Archimedes to ENIAC and Beyond” (2014).

All topics on Archimedes · ← The law of the lever · The quadrature of the parabola →

Image credits.