# 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.

Every point on a **circle**

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

You might remember that, for similar polygons, the ratio between corresponding sides is always constant. Something similar works for circles: the ratio between the *all circles*. It is always 3.14159… – a mysterious number called **Pi***π* for “p”. Pi has infinitely many decimal digits that go on forever without any specific pattern:

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 *r*, the circumference is

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:

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.

## 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 ${toWord(n1)} 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 ${toWord(n)} 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**

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,

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

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

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):

There even is a *Pi day* every year, which either falls on 14 March, because