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Circles and Pi
Cuplikan
Introduction
Degrees and Radians
Tangents, Chords and Arcs
The Circle Theorems
Cyclic Polygons
Spheres, Cones and Cylinders
Conic Sections

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Circles and PiIntroduction

Waktunya membaca: ~35 min

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 circle has the same distance from its center. This means that they can be drawn using a compass:

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 similar. You can prove that by showing how all circles can be matched up using simply translations and dilations:

You might remember that, for similar polygons, the ratio between corresponding sides is always constant. Something similar works for circles: the ratio between the circumference and the diameter is equal for all circles. It is always 3.14159… – a mysterious number called Pi, which is often written as the Greek letter 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 :

01234π

For a circle with diameter d, the circumference is C=π×d. Similarly, for a circle with radius r, the circumference is

C= .

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.

Triangle
Square
Pentagon
Circle

Circumference = 100, Area = ${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:

rπr

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 ${n1}, this shape starts to look more and more like a .

The height of the rectangle is equal to the of the circle. The width of the rectangle is equal to of the circle. (Notice how half of the wedges face down and half of them face up.)

Therefore the total area of the rectangle is approximately A=πr2.

r2πr

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 ${n2}, this shape starts to look more and more like a .

The height of the triangle is equal to the of the circle. The base of the triangle is equal to of the circle. Therefore the total area of the triangle is approximately

A=12base×height=πr2.

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:

A=πr2.

Calculating Pi

As you saw above, π=3.1415926… is not a simple integer, and its decimal digits go on forever, without any repeating pattern. Numbers with this property are called irrational numbers, and it means that π cannot be expressed as a simple fraction ab.

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 like a circle:

In 1665, Isaac Newton managed to calculate 15 digits. Today, we can use powerful computers to calculate the value of Pi to much higher accuracy.

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 International Space Station orbits Earth!

Of course, you don’t need to remember that many digits of Pi. In fact, the fraction 227=3.142… is a great approximation.

One approach for calculating Pi is using infinite sequences of numbers. Here is one example which was discovered by Gottfried Wilhelm Leibniz in 1676:

π=41−43+45−47+49−4+…

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.

3
.
1
4
1
5
9
2
6
5
3
5
8
9
7
9
3
2
3
8
4
6
2
6
4
3
3
8
3
2
7
9
5
0
2
8
8
4
1
9
7
1
6
9
3
9
9
3
7
5
1
0
5
8
2
0
9
7
4
9
4
4
5
9
2
3
0
7
8
1
6
4
0
6
2
8
6
2
0
8
9
9
8
6
2
8
0
3
4
8
2
5
3
4
2
1
1
7
0
6
7
9
…

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

Search for a string of digits:
3.

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 π≈3.14, or on 22 July, because π≈227.

Archie