How Pi was Estimated

Pi is a very significant constant in mathematics due to its many uses. Many people find it a magic number because of its infinite chain of random decimals; but it is quite simple and straight forward when we know how it was actually derived.

Pi was originally described as the ratio between the circumferences of a perfect circle to its diameter. Mathematicians would have been able to estimate this by getting an object with a circular perimeter and rolling it to find the complete distance it was traveled along its edge; then comparing this distance travelled to its diameter for a variety of differently sized objects. When this experiment was repeated it gave a decently precise value of a number a little greater than 3.1 or C=2πr. Of course, we have formulas to calculate it more accurately, but at the time they had to work with their limited measuring tools.

People of past generations also had another model of roughly finding Pi by using the area of a circle. Without the equations of our modern world, it is pretty difficult to find the area of a circle; the easiest way would be to get a volumetric displacement in water and divide by the object’s depth but that still leaves error due to their inferior measuring tools. So instead they estimated the area of a circle using an octagon with a diameter of 2. We know that these shapes if overlapped have decently similar areas so if we set the diameter to 2 we could find the area.

Before we run our calculations we should think why would we make our octagon with a diameter of 2 this is because if we did create this octagon with a diameter of 2 the radius would be 1 and we could eliminate this variable from calculation because anything times 1 is itself. So let’s find the area of the square encompassing the octagon; this would be 4 because it is side-one times side-two or 2*2=4. Then if we split this octagon into 9 smaller squares by cutting it into thirds vertically and horizontally, we can find the area of each part. Each full square has an area of 4/9 and since there are 5 of them those parts of the area total to 20/9. Then the diagonal half parts turn into half of a full square or 2/9; and since there are 4 of them those parts of the area total to 8/9. Then to find the total area of the octagon we add 20/9 + 8/9 = 28/9 = 3.11. We know Pi is 3.14159… but this is a pretty precise way of estimating the Pi variable. Then we can extrapolate this to find a consistent relationship of A=πr^2.

The first recorded use of Pi is within the structure of the Great Pyramid at Giza (2589-2566BC). It was built with a perimeter of 1760 cubits and a height of 280 cubits, which is a ratio of 1760/280 or roughly 2Pi. Some believe this to be of deliberate design while others believe this ratio was purely coincidental but either way this was the first record.

In our modern world we have computers that can calculate Pi to an almost infinite decimal, but it is still fascinating how mathematicians were able to derive all of these relationships such a long time ago. Hopefully after reading this article you can understand the general way mathematicians formulated Pi; but for a very visually and thorough description of the estimation of Pi I would suggest looking at this video.