Pi (π) is a mathematical constant defining the ratio between the circumference of a circle and its diameter. In other words, if you divide the circumference of a circle by its diameter, you always get the same number, which is approximately equal to 3.14159.
Discover FizziQ
How to measure it in class
FizziQ makes it possible to recover the value of π experimentally from physical measurements, without ever using the π key on the calculator. The most accessible method is that of the simple pendulum, whose period involves 2π.
Steps:
- Build a simple pendulum: a small dense mass hanging from a thin string. Measure the length ℓ of the string, from the attachment point to the center of the mass, with a tape measure.
- Pull the pendulum aside by less than 10° and release it without initial velocity: beyond that, the period is no longer independent of the amplitude and the formula no longer applies.
- Record the oscillations with FizziQ’s sound meter, producing a sound marker (a light contact) at each pass, or film the motion and mark it in the Kinematics module.
- Time 20 complete oscillations rather than a single one, then divide by 20: the triggering uncertainty is thus divided by twenty.
- Calculate π from the measured period: π = (T/2)·√(g/ℓ), with g = 9.81 m/s². Compare with 3.1416 and express the relative deviation.
- Repeat for three different lengths, plot T² as a function of ℓ and deduce π from the slope of the graph, which equals 4π²/g.
Scientific activities on this topic
Learn more
Irrational and transcendental: two distinct properties
These two words are constantly confused, even though they say different things. A number is irrational if it cannot be written as a fraction of two integers; its decimal expansion is then infinite and non-repeating. The irrationality of π was proved by Johann Heinrich Lambert in 1761. A number is transcendental if it is the root of no non-zero polynomial with integer coefficients. Transcendence is a stronger property: every transcendental number is irrational, but the converse is false. Thus √2 is irrational without being transcendental, since it is a root of x² − 2 = 0. The transcendence of π was established by Ferdinand von Lindemann in 1882.
Squaring the circle
Lindemann’s result closed a problem twenty-four centuries old. Constructing with straightedge and compass a square with the same area as a given disk would amount to constructing the number √π from the unit. Yet it is known that every number constructible with straightedge and compass is algebraic: it is the solution of a polynomial equation with integer coefficients. Since π is transcendental, √π is too, and is therefore not constructible. Squaring the circle is not difficult, it is impossible, and it is indeed transcendence - not mere irrationality - that establishes this.
Measuring π experimentally
Several protocols make it possible to obtain π without knowing it in advance. The most direct one consists of measuring the perimeter and diameter of circular objects of various sizes, then plotting the perimeter as a function of the diameter: the slope equals π. The Monte Carlo method relies on chance: points are drawn at random in a square of side 2R and the proportion of those falling inside the inscribed disk, equal to π/4, is counted. Its convergence is slow - the uncertainty decreases as 1/√N, so that the number of draws must be multiplied by one hundred to gain a single digit. The physical route, finally, goes through the period of a pendulum or of a mass-spring system, where π appears via the 2π factor. The article Measuring Pi with smartphone sensors: a fun and educational challenge details five of these approaches, from Vitruvius’ odometer to the Doppler effect in circular motion, via the Monte Carlo method.
Why π appears everywhere in physics
Pi emerges whenever a phenomenon has rotational symmetry or a periodic character. In circular kinematics, the angular velocity is written ω = 2πf and the linear velocity v = ωR: a complete turn equals 2π radians. Areas and volumes of revolution involve π: area of a disk A = πR², volume of a sphere V = (4/3)πR³, volume of a cylinder V = πR²h. In magnetism, the field created by an infinite straight wire is B = μ₀I/(2πR): the 2πR is simply the perimeter of the circle along which the field lines close. The 4π factor in Coulomb’s law, F = q₁q₂/(4πε₀r²), has the same geometric origin: it is the surface area of a sphere of unit radius. In quantum mechanics, finally, the reduced Planck constant ħ = h/(2π) appears in the indeterminacy relation ΔxΔp ≥ ħ/2, because the wave description always comes back to phases expressed in radians.
Orders of magnitude
π ≈ 3.141 592 653 589 793. The approximation 22/7 is accurate to 4 × 10⁻⁴, 355/113 to 3 × 10⁻⁷. Archimedes, around 250 BC, bracketed π between 223/71 and 22/7 by comparing inscribed and circumscribed polygons with 96 sides. Fifteen decimal places are enough to calculate the circumference of the Earth to the millimeter; NASA uses fifteen for its interplanetary navigation calculations, and thirty-nine would suffice to calculate the circumference of the observable universe to within the thickness of a hydrogen atom. Yet more than 10¹⁴ decimal places have been computed, essentially as a test of intensive computing.
Formula
Definition of π, the constant ratio of the perimeter of a circle to its diameter:
π = P / d
where:
- P: perimeter (circumference) of the circle (m)
- d: diameter of the circle (m)
Area of a disk and volume of a sphere:
A = πR² and V = (4/3)πR³
where:
- R: radius (m)
- A: area (m²)
- V: volume (m³)
Period of a simple pendulum at small oscillations, the relation that makes it possible to measure π:
T = 2π√(ℓ/g)
where:
- T: period of the oscillations (s)
- ℓ: length of the pendulum (m)
- g: gravitational acceleration, 9.81 m/s²
Monte Carlo method: for N points drawn at random in a square and n falling inside the inscribed quarter-disk,
π ≈ 4n / N
where:
- n: number of points located inside the disk
- N: total number of points drawn
Angular velocity of uniform circular motion:
ω = 2πf = 2π/T
where:
- ω: angular velocity (rad/s)
- f: frequency (Hz)
- T: period (s)
Application examples
-
A bicycle wheel 70 cm in diameter travels π × 0.70 ≈ 2.2 m per revolution: counting the revolutions is enough to measure a distance, which is the principle of the odometer.
-
A pendulum 1.00 m long has a period of 2π√(1.00/9.81) = 2.006 s. Conversely, timing this period gives π to better than 1 % with a simple string and a mass.
-
A Monte Carlo draw of 10,000 points typically gives π = 3.14 ± 0.03: the precision remains modest, which concretely illustrates the 1/√N convergence.
-
On a merry-go-round turning at 6 revolutions per minute, that is f = 0.10 Hz, the angular velocity is ω = 2π × 0.10 ≈ 0.63 rad/s.
-
A straight wire carrying 10 A creates at 5 cm a field B = (4π × 10⁻⁷ × 10)/(2π × 0.05) = 4.0 × 10⁻⁵ T, comparable to the Earth’s magnetic field.
-
A tennis ball 6.7 cm in diameter has a volume of (4/3)π × (0.0335)³ ≈ 1.6 × 10⁻⁴ m³, or about 160 mL.
FAQ
Q: What is the difference between irrational and transcendental? A: Irrational means “cannot be written as a fraction of integers”. Transcendental means “is the root of no polynomial equation with integer coefficients”. The second property implies the first but not the other way around: √2 is irrational and yet a root of x² − 2 = 0, hence algebraic and not transcendental. π, for its part, is both.
Q: Why can’t the circle be squared? A: Because the numbers constructible with straightedge and compass are all algebraic, whereas π is transcendental. Constructing a square with the same area as a disk would require constructing √π, which is therefore ruled out. Lindemann proved it in 1882; it is not a question of drawing technique but an impossibility in principle.
Q: How many decimal places of π are needed in practice? A: Very few. Three or four decimal places suffice for any lab calculation, where the uncertainty comes from the measurements and not from π. Fifteen decimal places already allow interplanetary trajectory calculations to the meter. The millions of computed decimal places serve to test algorithms and machines, not to do physics.
Q: Why does my measurement of π with the pendulum give 3.05? A: Most often because the amplitude was too large: beyond about ten degrees, the actual period exceeds 2π√(ℓ/g) and the calculated π comes out too small. The other classic causes are a length measured to the top of the mass instead of its center, and timing over a single oscillation.
Q: Why does π appear in formulas with no apparent circle, such as Heisenberg’s? A: Because these formulas describe wave-like or periodic phenomena. A phase is measured in radians, and a complete cycle equals 2π radians. In ΔxΔp ≥ ħ/2, the π is hidden in ħ = h/(2π): it comes from the passage from frequency to angular frequency, hence indeed from a rotation.
Related concepts
Uniform Circular Motion - Pendulum - Period - Frequency