# Pi Experiments Calculator

Created by Purnima Singh, PhD and Davide Borchia
Reviewed by Steven Wooding
Last updated: Jul 13, 2022

Pi, π, isn't just a number! It has always interested our species, has many intriguing properties, and is fundamental in physics, mathematics, and many other sciences. But after all, it's just neat! Most people know about the famous Pi Day, the annual celebration falling on the 14th of March, but did you know about the Pi Approximation Day?

On the 22nd of July, follow us on this adventure over the digits of pi! Learn why July 22 is Pi Approximation Day and how to celebrate it: we will start with a formal introduction (after all, we need to meet the host at this party), and from there, we will teach you why pi is so essential, and why we need it. And remember, this is not a lecture, but a party!

## 3. What is pi?

You ring the bell to pi's house: everything is a circle here, from the door to the walls, and you don't have the time to understand how a spherical house stands when a curious character opens the door. Meet π, our host.

Pi (π) is a mathematical constant that describes the ratio of the circumference $C$ of a circle to its diameter $d$. Simple, isn't it?

$\small \pi=\frac{C}{d}$

Pi has an interesting personality! Other numbers accuse it of being irrational, but it likes to say that it's all a matter of being transcendental: it's not like the other numbers! Let us explain what does that means.

Pi is an irrational number, which means that it is impossible to define it as the ratio of two integer numbers; on the other hand, the number $3.0656934306569343...$ is rational since it is the result of $\frac{420}{137}$ — you can see that there is a set of repeating digits.

Pi is also a transcendental number: there is no non-zero polynomial (of finite degree) with pi as root.

🙋 OK, the transcendental part was a bit complicated. Here's an example to make it easier: take $\sqrt{2}$. It is an irrational number (trust us!), but it is the root (not the mathematical operation) of the polynomial $x^2-2=0$. We can't find such an expression for pi!

We mentioned that pi is a mathematical constant: a specific number with interesting properties that often appear in mathematics. $\pi$, $e$ (the base of natural logarithms, $i$ (the complex unit), are examples of mathematical constants: they got their names to distinguish them from the "ordinary" numbers.

🔎 There is a never-ending war in science: which value of pi to use. Ask an experimental physicist, they'll say $3.14$; ask a theoretical physicist, and you'll hear "one", ask an engineer, they'll say "three". Do you want to start a fight? Bring them together and ask the same question – disclaimer: the author is a physicist!

## 1. Is there an end to pi?

Pi is introducing itself to the guests. But what an introduction: you wonder if it will ever end reading its value. 🙄

Bad news: the presentation won't end as the number of digits of pi is infinite!

The reason is that pi is an irrational number: if there were a pair of numbers with a ratio equalling pi, somewhere in the digits of pi, there would be a repeating sequence. Since no such pair exists, pi digits never repeat in the same order: we can compute an infinite amount of them!

🔎 Every number you can imagine exists somewhere in pi: from your date of birth to arbitrarily long sequences of the same digit. By the way, starting at the position $762$, you can find six 9s: $...07211349999998372978...$. Imagine how surprised the first person who saw that should have been!

## 4. Where can you find pi?

Pi is everywhere; we are not sure where the pie is, though. Apart from being a fundamental mathematical constant, pi keeps on appearing in other fields, from to geography, from biology to astronomy.

We want to prove to you that there is space for fun and curiosity, even when talking about mathematical constants!

Let's take a look at some of the places you can find pi.

#### Pi in math — maths in pie

Circles! ⭕

Pi is intimately connected to circles, being the ratio of circumference and diameter. Surprise, surprise? Wherever there is a rotation, pi appears: rotation solids like spheres, cones, and cylinders have their volume associated with pi.

And talking of rotations, trigonometry makes extensive use of pi: the measurement units of an angle are expressed in its multiples to make the use of functions like sine and cosine easier to use.

We then have that $180\degree$ corresponds to $\pi$ radians, while $360\degree$ corresponds to $2\pi$ radians — for every circular pie, you have then two pi.

#### Pi in physics

Let's return with our feet on the ground! In physics, pi usually appears when a sphere is introduced (spherical coordinates need pi: discover them with our spherical coordinates calculator) or when we study a harmonic motion or a wave.

We are not surprised to find pi in fluid dynamics or studying the atmosphere. Pi also appears when describing probabilistic phenomena — we will see this later!

#### Pi in biology — not only circles

Pi also appears where we don't expect it: biology. We tend to think of biology as a field far from math: it's not even considered a hard science (sometimes the distinction assumes negative characteristics: we disagree. Soft sciences are cool!) There are a in living things that contain pi: periodic events like heartbeat and biological rhythms, wave-like events like epidemics, and population dynamics.

Close your eyes and apply a little pressure on them with your hands. If you are doing it right, you will see geometrical patterns of light, changing and moving: those are phosphenes. Even if there is no definitive answer on their origin, some scientists suppose their pattern arises from equations .

## 1. Why do we celebrate Pi Approximation Day?

Now we are sure you got it: pi is essential. In 1988 at the Exploratorium in San Francisco, Larry Shaw decided to celebrate the mathematical constant on the day that resembles most of its digits: the 14th of march, 3-14: it was the birth of Pi Day!

But that's not it. Around (pun intended) the world, mathematicians complained about a date format used only in the U.S. (mm/dd/yyyy) and decided to create another Pi Day! But how? Allow us to introduce the approximations of pi.

If you are:

• Lazy;
• In a rush;
• A Babylonian;
• An engineer; or
• All four above (a lazy Babylonian engineer in a rush);

Then use the approximation $\pi=3$. We don't even need a ratio here. Now go away!

If a single decimal digit is enough, the ratio $25/8=3.125$ will be enough for you. This one is easy to calculate!

The first approximation that shows us the most famous approximation of pi, $3.14$, is $22/7$. We owe this one to Archimedes (the "Eureka" guy), that also proved the inequality:

$\frac{223}{71}<\pi<\frac{22}{7}$

To the surprise of no one, we remember more easily $22/7$, and... isn't it the 22nd of July?

That's it; we got our date! We can celebrate pi also on the date corresponding to the first approximation to give $3.14$!

🙋 Are you saying something about better approximations? Yes, there are, and they are really good; however, they are also clumsy: are you sure you want to use $9801/(2206 × √2) \approx 3.14159273$ instead of $22/7$?

## 5. How to celebrate Pi Approximation Day?

The way you want to celebrate Pi Approximation Day is entirely up to you!

Here are some ideas to celebrate Pi Approximation Day:

• Bake a pie (this is almost mandatory!);
• Memorize as many digits of pie as you can;
• Run in circles, or organize a circular parade!;
• Learn something about for fun; and
• Search for local celebrations: many museums have special programs for Pi Approximation Day.

However, if you want to celebrate the day at home, we can help you: let's make that pi-inspired pie and while it bakes, let's calculate some pi!

## 9. Pi Approximation Day quest: find pi using math

We will give you a method to find pi using mathematical tools: but we warn you: reaching a familiar value takes a lot of time!

Let's introduce the Leibniz series, bearing the name of one of the best mathematicians in history, not the cookies (we are not stopping you from eating a series of Leibniz, though).

The series is the expansion of the arctangent function. Arctangent... what is it? It's enough to know it is an inverse trigonometric function: circles again. But if you want to learn more about it, we've got your back with our arctan calculator! And expansion? In math, expansion means that we are dealing with a series of algebraic sums that eventually converge on the true value of the function.

$\small \begin{gather*} \arctan{x}= x\cdot\sum_{k=0}^{\infty}\frac{-1^k}{2\cdot k-1}\\\\ x-\frac{1}{3}x+\frac{1}{5}x-\frac{1}{7}x+\frac{1}{9}x+... \end{gather*}$

By knowing that $\arctan{1}= \frac{\pi}{4}$ we can find the value of $\pi$ by a simple substitution: $x=1$.

$\small \pi=4\cdot\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+...\! \right)$

Try to calculate the terms by hand: it will be satisfying! If you don't remember how to do it, we've got you covered: try our long division calculator. A good starter is calculating the first seven terms.

Summing allows us to estimate the value of $\pi$ as $3.33972$. That's almost a two-tenths difference, a $6.3$ percent error from the actual value, but we can say "close enough"!

To reach a good approximation, we need to iterate the series a high number of times. The first time the value of the series stops oscillating outside the range $3.14\pm0.01$ (thus assuming the most known value of pi definitively) is at step $625$: if you want to compute pi this way, you have to be patient!

## 2. Four experiments to celebrate Pi Approximation Day

Luckily, there are easier — and funnier — ways to find the value of pi! We are Omni Calculator, and our mission is to help you calculate everything you need in your daily life — but this time we decided to also have tons of fun!

Here are four ways to calculate pi: in our calculator, you will be able to insert your data and calculate your personal value of pi; here, we have fun explaining to you how those experiments work. Let's start with the first one!

#### Find pi the Egyptian style: measuring circular objects

This first method is the easiest way to find pi, and we said it: it is like cheating since we will use the definition of pi to calculate its value.

Take any circular object you want: a plate, a CD, an Oreo cookie, or the cake you just baked. Then you will need some high-tech measuring instruments: a string and a ruler.

Now take the two measurements you need: the diameter of the circle, and by wrapping the string around the object, and measuring the length of it, the circumference.

Calculate the ratio:

$\small\pi=\frac{C}{d}$

where:

• $C$ – Circumference; and
• $d$ – Diameter.

🔎 Is the result disappointingly far from the value of pi? Don't worry, and let's call in our help statistics: repeat the measurements, both on the same object and other things. The more, the merrier!

#### Measure pi using Archimedes intuition

Take a circular object you can cut (cardboard, pizza... no plates this time), and start cutting it in slices. Choose an even number of slices, like $4$, and start with a small number because we are going to do some more cutting, and we want your life to be easy!

Arrange your slices in such a way to have the outside facing alternately opposite directions. The resulting shape has two sides with length equal to the radius $r$ of the original circle, and the other two sides slightly "wavy", with length $\frac{C}{2}$, where $C$ is the circumference.

Repeat the cutting step, dividing each slice in two. Rearrange the now eight slices in the same alternated fashion. The shape you will get still has the same measurements, but the two irregular sides look less wavy.

You may see in which direction we are going! Repeat the steps again! Now you have $16$ slices, and the shape after the rearrangement will look pretty similar to a rectangle. It's math time then: while calculating, eat some pizza too!

Which are the measurements of the rectangle sides? The height will be $h=r$ while the base will be $b=\frac{C}{2}$. Knowing the formulas for the area and the circumference of a circle, we can write:

\small \begin{align*} A_{\text{rect}}&=A_{\text{circ}}\\\\ b\cdot h&= r^{2}\cdot \pi\\\\ \frac{C}{2}\cdot r&=\frac{2\cdot r\cdot \pi}{2}\cdot r \end{align*}

Rearrange the last equality, and you will find:

$\small\pi=\frac{C\cdot r}{2}$

Insert your measurements in our calculator, and let us do the math. 😉

🔎 This method allows you, as a side effect, to prove that the area of a circle is $A_{\text{circ}}=r^2 \cdot \pi$. This was the original proposition by Archimedes!

#### Find pi using statistics — and hot dogs

This method of finding pi is the most unexpected we propose here. To conduct this experiment, you will need some hot dogs, tape, and a bit of patience.

1. Clean the floor.
2. Take the hot dogs and measure their length.
3. Mark a certain number of equally spaced, parallel lines on the floor with the tape. The distance between lines must not be greater than the length of the hot dogs.
4. Face the lines perpendicularly.
5. Start throwing the hot dogs.
6. Keep throwing: the higher the number of hot dogs, the better.
7. Count the hot dogs that crossed one of the lines.

Before the math, we will give you an explanation. The original experiment was devised by George-Luis Leclerc, Comte de Buffon (a name that fills your mouth!), a French philosopher. Instead of using chien chauds, he proposed needles. But the experiment still holds; the only condition is that the objects used have the same length.

🔎 This simple physical experiment is an example of the Monte Carlo method, a class of algorithms that use random sampling to solve numerical problems. However, for more than two centuries since Buffon had this intuition, scientists had not been ready to use this method consistently: computers came in handy!

Now, time for some math and probabilities.

To demonstrate the experiment's validity, we will assume that the length of the hot dog $l$ is equal to the spacing between lines $x$. This will simplify things, but the general case is not much different.

We can define two variables to identify the position of a hot dog:

• The distance between the center of the hot dog and the closest line, $d$; and
• The angle at which the hot dog falls, with respect to the lines: $\theta$.

These two values have the following bounds:

• $D$ is never bigger than half the distance between lines: $0\leq D\leq\frac{x}{2}$;
• $0\leq \theta < 180\degree$.

A hot dog crosses a line if and only if:

$\small D\leq \frac{x}{2}\sin{\theta}$

Where $\sin{\theta}$ is the sine function of the angle. We now plot this curve as a function of the angle $\theta$.

The shaded area $A_{\text{hit}}$ below the curve is a hit: any hot dog which falls with an angle and a distance belonging to that area crosses one of the lines. The area of the dashed rectangle includes the hot dogs that don't cross any line.

It's time for some math: the value of the shaded area is:

$\small A_{\text{hit}}= \frac{x}{2}\cdot 2= x$

On the other hand, the area of the dashed rectangle is:

$\small A_{\text{all}}=\frac{x}{2}\cdot \pi$

Compute the ratio $A_{\text{all}} / A_{\text{hit}}$. This quantity is nothing else but the probability of a hot dog to cross one of the lines:

$\small \frac{A_{\text{hit}}}{A_{\text{all}}}=x\cdot \frac{2}{x\cdot \pi} = \frac{2}{\pi}$

Using a simplified interpretation of the Monte Carlo method, we can say that the areas correspond to the number of events, in this case, hot dog launches. That's how we can find pi: count the hot dogs that crossed the lines, $N_{\text{hit}}$, and with $N_{\text{all}}$ being the number of total launches, we have the following result.

$\small \frac{N_{\text{hit}}}{N_{\text{all}}}=\frac{2}{\pi} \rightarrow \pi=2\cdot\frac{N_{\text{all}}}{N_{\text{hit}}}$

⚠️ This result strongly depends on the values of $N_{\text{hit}}$ and $N_{\text{all}}$: to have a decent approximation of pi, those numbers must be rather high!

#### Finding pi with a pendulum

The last experiment we will introduce uses some basic experimental physics: you will build a simple pendulum!

What do you need?

• A string. The perfect length is $2-3$ feet (or $1\ \text{m}$, if you speak metric);
• A heavy object that you can easily tie to the string (why not a pie?);
• A ruler; and
• Your phone or any other stopwatch.

Tie the heavy object and secure one end of the string to it. Fasten the other end to the ruler (if there is a hole in it, use it!). Now tape the ruler to a surface that allows your freshly made pendulum to swing freely. Measure the length of the string from the ruler to the approximate center of mass of the object. You are all set.

The pendulum will sit with the string perpendicular to the ground: that's what physicists call the equilibrium state.

Keep your phone ready to measure the time, move the object a little from the static position, and release it — trying to be as delicate as possible, making it oscillate in a single direction. At the same time, start to measure the time.

Wait for the pendulum to complete ten oscillations, then pause the stopwatch. It's math time!

The formula for the period of a pendulum oscillation is:

$\small T=2\pi\sqrt{\frac{L}{g}}$

where:

• $T$Period of an oscillation;
• $L$Length of the string (you already measured it!); and
• $g$Standard gravitational parameter, with value $g=9.81\ \text{m}/\text{s}$.

You can imagine what to do now. The time you measured is equal to $t= 10\cdot T$: the calculations are easy!

$\small t= 10\cdot 2\pi\sqrt{\frac{L}{g}} \rightarrow \pi = \frac{t}{20}\sqrt{\frac{g}{L}}$

💡 Do you want to make things even easier? If you have space, build a pendulum with length $\thicksim 2.45\ \text{m}$. Now the square root can be approximated to $1/2$: $10$ oscillations would then take... $10\cdot \pi$. And while ten pies would be too much, this one is a nice result!

## 6. A final word

Pi has no end, but this article does! While the Earth rotates $2\pi\ \text{rad}$ around its axis, join us in this weird mathematical celebration. Bake a pi pie and have fun calculating the value of this constant with us!

Pi-pi! Oh, sorry: bye-bye!

Purnima Singh, PhD and Davide Borchia
Let's make mathematics enjoyable with pizza 🍕, hot dogs 🌭, and Oreos 🍪. Design experiments 🥼 using stuff from your kitchen and garage and calculate the value of Pi.
You'll have the most fun with the calculator if you do the experiments yourself.
Start by choosing your experiment 👨‍🔬👩‍🔬
Experiment
Using a pizza
🧠 This experiment is based on the proposition mentioned in the Archimedes' essay on the 'Measurement of a circle', which states that: The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.
Instructions:
1. Instead of making one right triangle out of a circle, we will make a rectangle which is a combination of two right triangles.
2. Take a pizza and cut it into thin slices.
3. Rearrange the pieces so as to make a rectangle.
4. Measure the length and width of the rectangle.
💡 You can also use a circular piece of cardboard or paper for this experiment.
Let's get that pizza! 🍕
Length
in
Width
in
Your calculated value of pi 🎉....
Pi (π)
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