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Approximating π

 2022-03-14 
A few weekends ago, I visited Houghton-le-Spring to spend two days helping with an attempt to compute the first 100 decimal places of π by hand. You can watch Matt Parker's video about our calculation to find out about our method and how many correct decimal places we achieved.
One of my calculations
Spending two days computing an approximation of π led me to wonder how accurate calculations using various approximations of π would be.
One nice way to visualise this is to ask: what is the largest circle whose area can be correctly computed to the nearest mm² when using a chosen approximation of π? In this blog post, I'll answer this question for a range of approximations of π.

3

First up, how about the least accurate approximation we could possibly use: π = 3.
Using this approximation, the areas of circles with a radius of up to 1.88mm could be calculated correctly to the nearest mm². That's a circle about the size of an ant.

Pi Day: 3.14

Today is Pi Day, as in the date format M.DD, today's date is the first three digits of π. Using this approximation, circles with a radius of up to 17.7mm or 1.77cm can be calculated correctly to the nearest mm². That's a circle about the size of my thumb.

Pi Approximation Day: 22/7

In the date format DD/M, 22 July gives an approximation of π that is more accurate than 3.14. Using this approximation, circles with a radius of up to 19.8mm or 1.98cm can be calculated correctly to the nearest mm². That's a slightly bigger circle that's still about the size of my thumb.

Our approximation

In Houghton-le-Spring, our final computed value was 3.1415926535886829815214... The first 11 decimal places of this are correct.
Using this approximation, circles with a radius of up to \(6.71\times10^5\)mm or 671m can be calculated correctly to the nearest mm². That's a circle about the size of Regent's park.

The 100 decimal places we were aiming for

If we'd avoided any mistakes in Hougton-le-Spring, we would've obtained the first 100 decimal places of π. Using the first 100 decimal places of π, circles with a radius of up to \(7.8\times10^9\)mm or 7800km can be calculated correctly to the nearest mm². That's a circle just bigger than the Earth.

The 527 decimal places that William Shanks computed

In 1873, William Shanks computed 707 decimal places of π in Houghton-le-Spring. His first 527 decimal places were correct. Using his approximation, circles with a radius of up to approximately \(10^{263}\)mm or \(10^{244}\) light years can be calculated correctly to the nearest mm². The observable universe is only around \(10^{10}\) light years wide.
That's a quite big circle.
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Comments

Comments in green were written by me. Comments in blue were not written by me.
When does "MM" give 14 for the month?
Steve Spivey
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I wonder if energy can be put into motion with pi, so that would be a lot of theoretical energy
Willem
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