Pi

Background

The history behind finding the value of pi is quite long...

Question

How was the value of pi discovered?

Answer

See below for the history of finding pi...

Analysis

Let's first talk about what exactly pi is. Pi is a Greek letter that represents the value we're talking about. It is an irrational number and so the closest we can come to expressing it as a decimal or a fraction is some sort of approximation.

It is defined as the circumference of a circle divided by its diameter:


Currently, school kids are taught to use either 3.14 or  as an approximation for pi. How was this number found?

Direct Measure

One way ancient cultures found an approximation for pi was to directly measure the diameter and circumference of larger and larger circles. The Babylonians, roughly 4000 years ago, found pi to be 3.125, or . The Egyptians, around 1650 BC, found the value for pi as 3.1605.

As a geometrical approximation

Another way to measure pi is to find it through calculating its area. Since we know how to calculate the area of a square, we can draw a square and inscribe a circle (put a circle in the centre such that it touches the square's perimeter) and within that inscribe another square, like this:



The circle is radius 1. The outside square is therefore equal to 4 square units and the inside square is equal to 2 square units. But we can get closer if we use more and more rounded shapes:



Here we've used an octagon. Clearly, the area of the octagon is closer to the area of the circle than the area of the square. And as we add more and more sides to the figure inscribed inside the circle, the closer it gets to being a circle.

It was known how to calculate the exact value of polygons (plane closed figures with equal straight lines, like a square and an octagon) and so as the number of faces, or sides, increased, the value of the polygon's area would more and more closely approximate the area of the circle.

Both the Greeks and the Chinese used this method. Archimedes narrowed the number down to . Zu Chongzi approximated pi as . To put these into decimal form:

Archimedes: 3.14084507042< pi < 3.142857
Zu Chongzi: 3.14159292035

More modern calculations have brought pi even closer to its actual value. There are a list of other methods used to refine pi in the wikipedia link.

Vocabulary used:

For more information check out these links (comment to add your favourite link):

This link, to exploratorium.com, has more on the history.
This link, to mathopenref.com, has more on the use of polygons to find pi.

Where might you have come from?

Fact-orials Index

Numbers:

• Irrational Numbers

Associated Operations:

• Estimation and Rounding

Geometry:

• Defining a Circle and Right Angels
• Squares

Where might we go?