Circles and PiIntroduction
For as long as humans have existed, we have looked to the sky and tried to explain life on Earth using the motion of stars, planets and the moon.
Ancient Greek astronomers were the first to discover that all celestial objects move on regular paths, called orbits. They believed that these orbits are always circular. After all, circles are the “most perfect” of all shapes: symmetric in every direction, and thus a fitting choice for the underlying order of our universe.

Earth is at the center of the Ptolemaic universe.
Every point on a A circle is the set of all points in two dimensions, at a fixed distance (the radius) from a given point (the center). A compass is a drawing tool used for creating circles or arcs. It consists of two arms – the needle on one end is placed in the center, while pencil on the other end traces out the curve.
There are three important measurements related to circles that you need to know:
- The radius is the distance from the center of a circle to its outer rim.
- The diameter is the distance between two opposite points on a circle. It goes through its center, and its length is
??? the radius. - The circumference (or perimeter) is the distance around a circle.
One important property of circles is that all circles are Two geometric shapes are similar if they have the same shape but a different size. We can move, rotate, reflect and resize one shape to match up with the other one. Their corresponding angles are equal and their corresponding sides are in the same ratio. A translation is a transformation that moves a figure in a specific direction, without changing its angle or shape. A dilation is a transformation that resizes a geometric shape, by making it bigger or smaller.
You might remember that, for similar polygons, the ratio between corresponding sides is always constant. Something similar works for circles: the ratio between the The circumference of a circle is the distance around its outside edge. A circle with radius r has circumference The diameter of a circle is the distance between two opposite points on its circumference, and passing through its center. Pi, often written as the Greek letter π, is the ratio of the circumference and the diameter of a circle. It is a transcendental number, and its value is approximately 3.14159265…
Here is a wheel with diameter 1. As you “unroll” the circumference, you can see that its length is exactly
For a circle with diameter d, the circumference is The radius of a circle is the distance (or a line segment) between its center and any point on its circumference.
Circles are perfectly symmetric, and they don’t have any “weak points” like the corners of a polygon. This is one of the reasons why they can be found everywhere in nature:

Flowers

Planets

Trees

Fruit

Soap Bubbles
And there are so many other examples: from rainbows to water ripples. Can you think of anything else?
It also turns out that a circle is the shape with the largest area for a given circumference. For example, if you have a rope of length 100 m, you can use it to enclose the largest space if you form a circle (rather than other shapes like a rectangle or triangle).
In nature, objects like water drops or air bubbles can save energy by becoming spherical, and minimising their surface area for a given volume.
Circumference = 100, Area =
The Area of a Circle
But how do we actually calculate the area of a circle? Let’s try the same technique we used for finding the area of quadrilaterals: we cut the shape into multiple different parts, and then rearrange them into a different shape we already know the area of (e.g. a rectangle or a triangle).
The only difference is that, because circles are curved, we have to use some approximations:
Here you can see a circle divided into wedges. Move the slider, to line up the wedges in one row.
If we increase the number of wedges to
The height of the rectangle is equal to the
Therefore the total area of the rectangle is approximately
Here you can see a circle divided into rings. Like before, you can move the slider to “uncurl” the rings.
If we increase the number of rings to
The height of the triangle is equal to the
If we could use infinitely many rings or wedges, the approximations above would be perfect – and they both give us the same formula for the area of a circle:
Calculating Pi
As you saw above, Irrational numbers are numbers that cannot be expressed as fractions of integers (rational numbers). For example, 0.333333… =
It also means that we can never write down all the digits of Pi – after all, there are infinitely many. Ancient Greek and Chinese mathematicians calculated the first four decimal digits of Pi by approximating circles using regular polygons. Notice how, as you add more sides, the polygon starts to look

In 1665, Sir Isaac Newton (1642 – 1726) was an English physicist, mathematician, and astronomer, and one of the most influential scientists of all time. He was a professor at Cambridge University, and president of the Royal Society in London. In his book Principia Mathematica, Newton formulated the laws of motion and gravity, which laid the foundations for classical physics and dominated our view of the universe for the next three centuries. Among many other things, Newton was one of the inventors of calculus, built the first reflecting telescope, calculated the speed of sound, studied the motion of fluids, and developed a theory of colour based on how prisms split sunlight into a rainbow-coloured spectrum.
The current record is 31.4 trillion digits. A printed book containing all these digits would be approximately 400 km thick – that’s the height at which the The International Space Station (ISS) was launched in 1998 and orbits Earth at an altitude of approximately 400 km. It is the largest human-made body in low Earth orbit, and has been inhabited continuously since launch. It can be seen with the naked eye from Earth.
Of course, you don’t need to remember that many digits of Pi. In fact, the fraction
One approach for calculating Pi is using infinite sequences of numbers. Here is one example which was discovered by Gottfried Wilhelm Leibniz (1646 – 1716) was a German mathematician and philosopher. Among many other achievements, he was one of the inventors of calculus, and created some of the first mechanical calculators. Leibniz believed that our universe is the “best possible universe” that God could have created, while allowing us to have a free will. He was a great advocate of rationalism, and also made contributions to physics, medicine, linguistics, law, history, and many other subjects.
As we calculate more and more terms of this series, always following the same pattern, the result will get closer and closer to Pi.
Many mathematicians believe that Pi has an even more curious property: that it is a normal number. This means that the digits from 0 to 9 appear completely at random, as if nature had rolled a 10-sided dice infinitely many times, to determine the value of Pi.
Here you can see the first 100 digits of Pi. Move over some of the cells, to see how the digits are distributed.
If Pi is normal, it means that you can think of any string of digits, and it will appear somewhere in its digits. Here you can search the first one million digits of Pi – do they contain your birthday?
One Million Digits of Pi
We could even convert an entire book, like Harry Potter, into a very long string of digits (a = 01, b = 02, and so on). If Pi is normal, this string will appear somewhere in its digits – but it would take millions of years to calculate enough digits to find it.
Pi is easy to understand, but of fundamental importance in science and mathematics. That might be a reason why Pi has become unusually popular in our culture (at least, compared to other topics of mathematics):
Pi is the secret combination for the tablet in “Night at the Museum 2”.
Professor Frink (“Simpsons”) silences a room of scientists by saying that Pi equals 3.
Spock (“Star Trek”) disables an evil computer by asking it to calculate the last digit of Pi.
There even is a Pi day every year, which either falls on 14 March, because
