mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

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.
×3      ×3      ×3      ×3      ×3
(Click on one of these icons to react to this blog post)

You might also enjoy...

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
×3   ×4   ×4   ×3   ×4     Reply
I wonder if energy can be put into motion with pi, so that would be a lot of theoretical energy
Willem
×3   ×3   ×3   ×3   ×4     Reply
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li> <logo>
To prove you are not a spam bot, please type "decagon" in the box below (case sensitive):

Archive

Show me a random blog post
 2025 

Jan 2025

Christmas (2024) is over
Friendly squares
 2024 
▼ show ▼
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

data graphs pi approximation day convergence asteroids sport estimation game show probability dragon curves fence posts folding tube maps tennis quadrilaterals manchester error bars 24 hour maths craft gaussian elimination chalkdust magazine pi runge's phenomenon triangles friendly squares christmas graph theory dataset php bodmas ternary chess puzzles nine men's morris matt parker binary inline code big internet math-off light advent calendar flexagons london underground fonts matrix multiplication tmip ucl game of life a gamut of games accuracy the aperiodical crossnumber hexapawn royal baby hats menace coins stirling numbers bots regular expressions cambridge chebyshev sound exponential growth go talking maths in public errors harriss spiral reddit zines logo stickers gerry anderson frobel phd numerical analysis mathslogicbot mathsteroids folding paper cross stitch datasaurus dozen final fantasy map projections edinburgh matrices games simultaneous equations machine learning draughts golden ratio radio 4 recursion pythagoras raspberry pi logs countdown guest posts programming matrix of minors geometry polynomials weather station pac-man palindromes youtube realhats bempp databet inverse matrices weak imposition national lottery signorini conditions finite element method squares arithmetic london royal institution bubble bobble interpolation books braiding world cup gather town newcastle finite group sorting oeis news python people maths captain scarlet plastic ratio crochet golden spiral rugby correlation mathsjam latex kings wave scattering european cup speed curvature probability data visualisation turtles trigonometry pascal's triangle dates misleading statistics propositional calculus hyperbolic surfaces football platonic solids javascript sobolev spaces pizza cutting wool dinosaurs statistics logic mean christmas card fractals martin gardner hannah fry boundary element methods preconditioning video games computational complexity manchester science festival matrix of cofactors numbers rhombicuboctahedron electromagnetic field approximation noughts and crosses anscombe's quartet standard deviation reuleaux polygons live stream geogebra determinants

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2025