The Best Approximation of Pi
It won’t surprise you to hear that every decimal representation of Pi is an approximation. This is because Pi is an irrational number that can’t be precisely expressed as a ratio of two integers. It is a non-terminating, non-repeating number. The current number of correctly calculated significant digits of Pi is more than 202 trillion [1, 2].
Given that every decimal representation of Pi is an approximation, it’s worth asking:
Which approximation should we use when designing things?
Common Pi approximations are:
Pi = 3
Pi = 3.14
Pi = 3.141519
Pi = 3.14151926
Pi = 355/113 = 3.1415929 [3]
Pi = 22/7 = 3.14285714 [3]
Pi = 3.15
This article explores the impact of Pi approximations on design decisions.
Why have a discussion about Pi?
Pi is the ratio of a circle’s circumference to its diameter. Thus Pi and its approximation is of concern to anyone who deals with circular geometry or circular movement [4]. This includes at least architects, engineers, designers, physicists, astronomers, and artists.
Pi is also a key constant used in the probability density function (PDF) calculation for normal distributions – the most commonly assumed distribution of variation [5]. As such, Pi and its approximation affect numerous other professions that involve statistics, including doctors, politicians, financiers, marketers, and more.
Pi and its approximation penetrate numerous facets of life.
Pi is not 3, or is it?
An engineering professor of mine would often say Pi is 3, and then proceed to calculate something using Pi=3, like the area moment of inertia of a beam with a circular cross section. I=Pi*D^4/64
Let’s be clear: he’s wrong.
Pi is not 3,
nor is it 3.1415926,
nor is it 3.14159265358979323846264338327950288419716939937510
582097494459230781
What he meant was, let’s approximate Pi as 3.
I once described this experience to a mathematics professor and she essentially came unglued. “Why would you accept a 4.5% error when you know the actual value of Pi?” she said.
Touché.
This argument has its flaws, though: Why would you use 3.14 as a Pi approximation and accept 0.046% error when you can use 3.14159 and only have to accept 0.0000891% error? Thus the endless pursuit of more Pi precision and the discovery of more than 202 trillion significant digits for Pi.
So why did my engineering professor approximate Pi as 3? There are at least two plausible reasons: for convenience and/or for safety.
Pi = 3 for convenience
If you are computing a value using mental math, Pi approximated as 3 is convenient. For basic quick calculations this may be justified as a way of knowing if a higher fidelity representation is warranted. For example, if the quick mental math yields values nowhere close to what you need for your design, it will not merit the creation of a spreadsheet or Python code where you can compute results with better approximations of Pi. In this case, quickly learning about the system behavior may be more valuable than reducing error below 4.5%.
Pi = 3 for safety
When dealing with real products and systems, many parameters (including constants) are used to compute system performance. Because of uncertainties in some parameters (such as the actual manufactured length of a rod), as well as uncertainties in how accurate our engineering models are (some models ignore friction, for example, even though it’s always present), engineers often use safety factors.
Safety factors are buffers to allow for uncertainty in numerous contributing factors. Safety factors typically range from 1.1 to 2. For example, if a mechanical component needs to withstand 100 kg of loading, it is common for an engineer to design it to withstand 150 kg, thus building in a safety factor of 1.5.
My professor could have approximated Pi as 3 because of how it “used up” or “contributed to” the buffer represented by the safety factor.
When choosing an approximation for Pi, a conscientious designer will understand what is being computed, the role that Pi plays in that computation, and whether the computed value would end up as more or less conservative based on the choice of the Pi approximation.
When my professor first said Pi is 3, it caught my attention, so I can remember the exact problem he was solving, even though it was 30 years ago. He was calculating the force required to shear a bolt, holding two plates together as shown below.
With Pi in the denominator of the shear stress equation, we can see that a smaller representation of Pi would result in a larger shear stress estimation. Given that the shearing of the bolt will happen when the calculated shear stress exceeds the shear strength, approximating Pi as 3, produces a safer, more conservative estimate of how much force (F) can be applied before the system fails. In this case, approximating Pi as 3 adds to the safety factor buffer.
A different example illustrates that the designer must understand the system and the equations used to estimate system performance when choosing how to represent Pi. Imagine you’re planning to pour a circular concrete pad and want to estimate the volume of concrete needed. If you approximate Pi as 3, you will underestimate the amount of concrete needed by 4.5%, which is clearly bad; it would be better in this case to approximate Pi as 3.15 and overestimate the amount of concrete needed.
If you are launching a spacecraft that you hope will intersect Mars on its solar orbit, you will want to predict Mars’ location as precisely as possible without over or under estimating. In this case, you would want the most precise approximation of Pi you can afford to compute.
What Pi’s approximation means to most designers
Our decimal representation of Pi is an approximation and we should be okay with that. We use approximations all the time in our lives. You express your age as an approximation. You state the speed at which you drive as an approximation. Meetings generally start at approximately a particular time. Babe Ruth’s longest home run is approximately 575 feet. This project will cost approximately $20,000. And so on.
Approximations are a natural part of design and everyday operation. In all cases, regarding Pi or not, designers should choose approximations that suit their design needs, which is typically about what information is needed and how many resources have to be spent to get it. Balancing this is an important job of the designer.
Consider the image of the horse below [6]. Under what conditions is it desirable to represent the horse as the sphere on the left, versus the collection of spheres to the right? Keep in mind that no matter how many spheres are added to the model on the far right, it is always only an approximation. As long as that approximation is adequate, additional effort expended for greater precision is not worth it.
Even though it is more uncomfortable to do, the same kind of story can be told by putting Pi=3 on the far left and Pi=3.1415926 on the right.
Archimedes first approximated Pi by considering the areas of polygons inscribed and circumscribed around a circle as shown below [7]. The consideration of both inscribed and circumscribed polygons gave lower and upper bounds of area and therefore Pi approximations [8].
Approximating a circle as a 12-sided polygon (shown above), yields the same area as if the area of a circle was calculated using Pi = 3. Adding additional sides to the polygon has this effect:
So we’re back to where we started, which approximation of Pi should I use if I am designing something?
Consider the following geometry. If a geometric visualization of the cylinder was the only consideration, Pi approximated at 3.14000 looks pretty good to me. That’s 0.05% error and represents the right-most point on the plot above (N=114). What does this tell me? It tells me that my typical use of Pi=3.14159 is likely overkill; that would be a polygon with 2820 sides.
Closing Facts (summary)
No matter what, you are using an approximation of Pi.
Whether you know it or not, you are choosing an approximation of Pi every time you use it in a calculation.
The best designers understand the role of approximations in decision making.
Thoughtful designers choose approximations that will provide them with an adequate understanding while expending the least amount of resources (human, time, financial); thoughtful designers will not waste resources.
The best approximation of Pi, for you to use, is the one that provides you with adequate information at the lowest cost in terms of human, time, and/or financial resources.
References
[1] Wikipedia, “Chronology of computation of Pi,” https://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80, accessed 10 March 2025.
[2] Williams, W., “They did it again: Tech publisher keeps on breaking Pi Calculation World Record — they almost doubled the previous one, reaching 202 trillion digits in 100 days and used 1.5PB of SSD storage,” Tech Radar, 5 July 2024, https://www.techradar.com/pro/they-did-it-again-tech-publisher-keeps-on-breaking-pi-calculation-world-record-they-almost-doubled-the-previous-one-reaching-202-trillion-digits-in-100-days-and-used-15pb-of-ssd-storage, accessed 10 March 2025.
[3] Wikipedia, “Milü,” https://en.wikipedia.org/wiki/Mil%C3%BC, accessed 10 March 2025.
[4] Wikipedia, “List of formulae involving Pi,” https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80, accessed 10 March 2025.
[5] Wikipedia, “Normal Distribution,” https://en.wikipedia.org/wiki/Normal_distribution, accessed 10 March 2025.
[6] Mattson, C, and Sorensen, C., Product Development: Principles and Tools for Creating Desirable and Transferable Designs, Springer Nature, Cham, 2020.
[7] Exploratorium, “A Brief History of Pi,” https://www.exploratorium.edu/pi/history-of-pi#:~:text=Archimedes%20knew%20that%20he%20had,brilliant%20Chinese%20mathematician%20and%20astronomer, accessed 10 March 2025.
[8] Wikipedia, ”Pi,” https://en.wikipedia.org/wiki/Pi, accessed 10 March 2025.
To cite this article:
Mattson, Chris. “The Best Approximation of Pi.” The BYU Design Review, 10 Mar. 2025, https://www.designreview.byu.edu/collections/the-best-approximation-of-pi.
