Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Happy pi day 2016!

      14 March, 2016
      It's March 14th, which in the US is written as 3/14 — and since 3.14 are the first three digits of that most famous of mathematical constants, π, today is celebrated internationally as pi day.
      Coin rolling

      How many revolutions will the smaller circle make when rolling around the bigger one?

      The number π is the ratio of the circumference of a circle to its diameter. To celebrate this lovely number, here's a little puzzle to ponder. Imagine a circle with radius 1 cm rolling completely along the circumference of a circle with radius 4 cm. How many rotations does the smaller circle make? The circumference of a circle with radius r is 2πr, so the circumference of a circle with radius 4r would be 8πr. Since 8πr÷2πr=4, it seems the answer must be four revolutions. But that's not true! The answer is actually 5! Can you figure out why?

      We found out about this curious question from Yutaka Nishiyama, a mathematical friend in Japan. You can read about the answer in his article Circles rolling on circles. Happy puzzling!

      Read more about...
      geometry
      circle
      • Log in or register to post comments

      math.nights

      14 March 2016

      Permalink

      To Arabic: https://goo.gl/l3Mt0L

      • Log in or register to post comments

      Anonymous

      15 March 2016

      In reply to Translation by math.nights

      Permalink

      Thank you for translation in Arabic.
      Yutaka Nishiyama

      • Log in or register to post comments

      Anonymous

      17 March 2016

      Permalink

      The additional rotation occurs because the smaller circle traverses the circumference of the larger circle, in the same way that the moon rotates once every time it orbits the Earth, even though it maintains the same face towards the Earth. Of course, this assumes that the smaller circle moves as well as rotates, and that the larger circle neither moves nor rotates. If the centres of the two circles are fixed and both are free to rotate, as gears, then the answer will be 4, not 5. Furthermore, if your frame of reference is the larger circle and that circle is allowed to both move and rotate, then it is possible to envisage the smaller circle appearing to rotate 4 times without it moving or rotating at all! - Paul Baron

      • Log in or register to post comments

      Anonymous

      3 April 2016

      Permalink

      The question without the accompanying diagram could also mean to roll the small circle within the larger one. If this is done you actually lose a rotation. The answer then is 3!
      The easiest way to calculate the rotations is to use the path taken by the centre of the small circle.
      If r[1] = radius of small circle; r[2] = radius of path outside and r[3] = radius of path inside the large circle then:-
      Rotations outside = (2 x pi x r[2])/(2 x pi x r[1]) = (2 x pi x 5)/(2 x pi x 1) = 5/1 = 5.
      Rotations inside = (2 x pi x r[3])/(2 x pi x r[1]) = (2 x pi x 3)/(2 x pi x 1) = 3/1 = 3.

      K. Selby

      • Log in or register to post comments

      Read more about...

      geometry
      circle
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms