The Greek letter 𝜋 ("Pi", pronounced - at least in the UK - like "pie") when encountered in mathematics is usually used as shorthand for a pretty important number. This number can be defined in various ways, but arguably the simplest is "the ratio of a circle's circumference to its diameter." To put it another way, 𝜋 tells you how many times bigger the circumference is than the diameter; or it's the number you get if you divide a circle's circumference by its diameter:
\[ \pi = \frac{C}{d} \]
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| Circumference, Diameter & Pi by T.Briggs is licensed under CC BY-SA 4.0 |
One way that 𝜋 could be calculated is by measuring the diameter and circumference of a given circle and dividing the latter by the former. But one of the reasons we might want to know a value for 𝜋 in the first place is that it is difficult to measure the circumference of a circle.
So how was an accurate value for 𝜋 first calculated?
Values for 𝜋 have been recorded throughout antiquity as being about 3* but the earliest surviving description (from around 250 BCE) of a method for finding a value for 𝜋 to arbitrary levels of accuracy** is attributed to ancient Greek Archimedes (yes, he of "Eureka!" fame).
To replicate Archimedes method (or algorithm) you just need to know a bit about regular polygons. Two things, specifically...
1. What is a regular polygon?
A polygon is a closed two-dimensional shape with any number of straight sides. Polygons that you're probably quite familiar with are triangles (polygons with 3 sides), quadrilaterals (polygons with 4 sides, including squares and rectangles), pentagons (polygons with 5 sides) and hexagons (polygons with 6 sides). The names of the polygons are constructed by appending a form of the greek word for the number of sides with "agon".***
A regular polygon is a polygon in which all sides are of equal length and all interior angles are of equal size.
and...
2. How can you find the perimeter of a regular polygon?
As all sides in any given regular polygon are equal, you could find the length of one of these sides and multiply it by the number of sides. To find one side you could measure it, but that assumes you've already constructed one of the appropriate size. Another method is described in the diagram below****.
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| Polygons & Their Perimeters by T.Briggs is licensed under CC BY-SA 4.0 |
Now that's out of the way we can look at...
Archimedes' algorithm for calculating ever more accurate estimates for 𝜋
Given that we can do the above with relative ease the algorithm itself is pretty easy to understand, if long-winded to perform. It rests on the idea that a polygon is kind of a circle if you think about it in the right way, so we can use a polygon of the right size to approximate a circle of a particular size.
We could start by drawing a circle with a diameter that we have chosen. It doesn't know what that diameter is, but we need to know it with decent accuracy. If we now draw an equilateral triangle such that the vertices of that triangle are all on the circumference (this is known as "inscribing" the triangle) we know one thing for certain: the perimeter of the triangle is less than the circumference of the circle: We have a lower bound for the circumference of the circle!
It is important to note here (and you might also like to refer back to the previous diagram to recall why) that the distance between the centre of the inscribed triangle and each of its vertices is the same as the radius of the circle.
Now we can draw a "circumscribed" triangle; that is an equilateral triangle drawn around the circle such that the centre of each of its sides form tangents to the circle (i.e. each side touches the circle at exactly one point). We know one thing for certain about this triangle: its perimeter is definitely bigger than the circumference of our circle: We have an upper bound for the circumference of the circle!
It is important to note here (and you might also like to refer back to the previous diagram to recall why) that the distance between the centre of the circumscribed triangle and the midpoint of each of its sides is exactly the same as the radius of the circle.
We have an upper bound and a lower bound for the circumference of the circle, and we can use this to calculate upper and lower bounds for 𝜋. We could take the average of these two values to quote as our estimate for 𝜋. It's a relatively large range, though, which means our estimate wouldn't necessarily be that close to the true value. Can we make it any more accurate?
Of course! Have a look at the diagram below. The situation I've just described, a circle with an equilateral triangle inscribed and another circumscribed, is shown in the top-left of the diagram. Examples of similar situations involving polygons with increasing numbers of sides are also included. Hopefully you can see that as the number of sides increases the inscribed and circumscribed polygons become closer in size, and both get closer to the size of the circle between them.
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| Approximating Circles with Polygons by T.Briggs is licensed under CC BY-SA 4.0 |
For regular polygons with more sides the perimeter of the smaller one gets larger and the perimeter of the larger ones get smaller, with both converging on the circumference of the circle. The upper and lower bounds calculated become closer to each other, allowing for a more accurate calculation for 𝜋 with each one.
Archimedes didn't follow exactly this process: he started with hexagons, calculating the perimeters of both the inscribed and circumscribed ones, and then dividing each by the diameter of the circle he had chosen. He then doubled the number, skipping straight to dodecagons. He got as far as a 96-sided regular polygon***** before, I assume, getting bored and heading off for a bath******. With 96 sides at his disposal he determined the lower bound to be $\frac{221}{71} \approx 3.1408$ and the upper bound to be $\frac{22}{7} \approx 3.1429$. The average of these two values is $\approx 3.14185$, which is correct up to the third decimal place.
Not bad!
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