Wagon Wheel Approximation of Pi

Let there be spokes of a wagon wheel. We only need to deal with the upper half of a unit circle. Divide the semicircle vertically into evenly spaced segments using bars.
The intersections of the spokes with the corresponding vertical bars creates truncated spokes of length , , rather than of length 1. Here and .
Notice that is undefined by an intersection because both the spoke and bar are vertical and coincide. The value of must be inferred from the pattern created by connecting the intersections—amazingly, approaches as increases!



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


This Demonstration leads to one of the many beautiful expressions for :
(from the French mathematician Viète, 1593).
This formula comes from the cosine half-angle formula, which yields
, and so on.
Interpolation using rather than greatly improves the approximation.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+