What’s so special about Pi?

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Formally, Pi is a mathematical constant whose value is the ratio of a circle’s circumference to its diameter (Pi = C/d). It’s also the same ratio of a circle’s area to its radius (Pi = A/r^2).

Pi is an irrational number, meaning it cannot be referenced exactly as a fraction, but only as a decimal which never ends or repeats (ie, it’s “transcendental). In fact, the current world record has Pi at a value with over a trillion decimal points of accuracy. This level of accuracy is extreme… in a practical sense, a physicist needs only 39 digits of Pi to make a circle the size of the universe accurate to the size of a hydrogen atom.

Pi = 3.1415926535897932384626433832….

Visualizing Pi

To help visualize the value of Pi, and where the “3.14159…” comes from, look closely at the animation below, courtesy John Reid and Arpad Horvath. Note that a point is marked along the circumference of the circle where the circle touches the line. As you roll the circle along a straight line, Pi (or 3.1415926….) is the spot where that marker touches the line again. No matter what diameter, the ratio will remain the same: Pi

circle unrolling, pi

One of the reason’s ? is special is because it is involved in many common formulae, such as the area of a circle (Pi*r^2) and volume of a sphere (4/3 Pi*r^3). From an early age, we learn that for any circle with radius r and diameter d=2r, the circumference is Pi*d and the area is Pi*r^2.

Practical Applications of Pi

Pi is everywhere in science, physics, mathematics, and engineering. In addition, the periods of all the trigonometric functions are either equal to Pi or 2Pi, and in statistics as part of the normal distribution (square root of Pi).

area of a circle, piJill Britton, educator and mathematician, helps to explain how Pi is used to derive the formula for the area of a circle.

To calculate the area of a circle with radius r, we cut a circle into 4 equal wedges as shown in the picture. We arrange the four wedges in a row, alternating the tips up and down to form a shape that resembles a parallelogram.

The reason for changing a circle into a “parallelogram” is because we don’t know how to calculate the area of a circle yet, so we transform a circle into a shape whose area we know how to calculate. As shown, the length of the bumped base (top or bottom) is equal to half of the circumference of the original circle and the length of the other side is equal to the radius r. During this process, no area has been lost or gained so that the area of this newly formed “parallelogram” is the same as that of the original circle. However, this “parallelogram” has bumps on both its top and bottom, so we still don’t know how to calculate its area.

area of a circle piTo solve this problem, the original circle is divided into a greater number of equal wedges. As we increase the number of wedges, the bumps become smoother and the parallelogram looks more and more like a rectangle (and we know how to calculate the area of a rectangle, it’s just Length X Height). As the number of wedges approaches infinity, the bumped “parallelogram” becomes a perfect rectangle, with its width equal to Pi and its height equal to r.

As illustrated above, the width of this newly formed rectangle equals half of the circumference of the original circle and the height is equal to the radius r. The formula for the area of a circle is then:

Area of a Circle = Area of the Estimated Rectangle With An Infinite Number of Wedges
Length X Width = (Pi * r) * r = Pi*

Wikipedia lists many (if not all) the formulas that involve Pi.  It is so ingrained in popular culture that a movie (Pi) was created in 1998, about a paranoid mathematician for a key number that will unlock the universal patterns found in nature.

Skeptical spoiler alert:  there’s no such pattern.

But Pi is pretty cool anyways.