Wrap a string around a can and then stretch it over the diameter: it fits just over three times. This goes for any can, wheel or coin. This “just over three” is the number π.

Who discovered

That the outline of a circle is a little larger than three diameters was already known by the Babylonians and Egyptians almost 4 000 years ago; they used values ​​like 3, $3\tfrac{1}{8}$ and 3,16. The first calculated π with certainty was Archimedes of Syracuse (3rd century BC): using polygons with 96 sides, he proved that π is between $3\tfrac{10}{71}$ and $3\tfrac{1}{7}$. The symbol π was used by the Welshman William Jones in 1706 and popularized by Leonhard Euler.

The story

Archimedes surrounded the circle by two polygons, one inside and one outside, and doubled the number of sides four times: 6, 12, 24, 48 and 96. The perimeters of the polygons approach the contour from below and above.

In the 5th century, the Chinese Zu Chongzhi reached 3,1415926 < π < 3,1415927, a record that lasted almost a thousand years. Today, trillions of decimal places of π are known, calculated by computer — but for almost everything in engineering, 3,14159 is enough.

In everyday life

  • Wheels and odometer. With each revolution, a wheel travels its length $C = 2\pi r$; the car dashboard counts the laps.
  • Bicycle. A wheel with rim 29 is about 74 cm in diameter with the tire: it travels about 2,3 m per turn.
  • Athletics tracks. The outside lanes start further ahead because their curve is longer.
  • Seamstress’ measuring tape. Measuring the waist is measuring a “circumference”.

The formula

Choose the radius and roll the wheel one turn. The unfolded red section is the circumference; the yellow marks are diameters.

TeoremaCircumference length

The ratio between the length $C$ of any circle and its diameter $d$ is always the same number, $\pi \approx 3{,}14159$. Like this,

$$C = \pi \cdot d = 2\pi r.$$

PropriedadeLength of an arc

An arc of $\alpha$ degrees is the fraction $\tfrac{\alpha}{360}$ of the circumference:

$$\ell = \frac{\alpha}{360^\circ} \cdot 2\pi r.$$

The demonstration that the ratio $C/d$ is the same for all circles uses similarity of polygons and is in the circle length chapter from high school.

Exercises

1

Calculate the length of a circle of radius 10 cm (use $\pi \approx 3{,}14$).

Ver resposta

$C = 2 \cdot 3{,}14 \cdot 10 = 62{,}8\ \text{cm}$

2

The outline of a round table measures 157 cm. What is the diameter of the table?

Ver resposta

$d = \dfrac{C}{\pi} = \dfrac{157}{3{,}14} = 50\ \text{cm}$

3

A wheel of 70 cm in diameter travels 1 km. Approximately how many turns does it make?

Ver resposta

Each lap: $C = 3{,}14 \cdot 0{,}7 \approx 2{,}2\ \text{m}$. Turns: $1\,000 \div 2{,}2 \approx 455$.

4

What is the length of an arc of 90° in a circle of radius 12 cm?

Ver resposta

$\ell = \dfrac{90}{360} \cdot 2 \cdot 3{,}14 \cdot 12 = 18{,}84\ \text{cm}$

True or false?

  1. The contour of any circle measures just over 3 diameters.
  2. π is exactly 3,14.
  3. By doubling the radius, the length of the circle doubles.
  4. An arc of 180° measures 2πr.

Sources: Archimedes (MacTutor) · Zu Chongzhi (MacTutor) · History of π (MacTutor).