DIY: Enrico Fermi’s Back of the Envelope Calculations


This post is part of Issue#16: Mathemagic

Using simplified assumptions to do rough calculations by noting them down on any available scrap of paper such as the back of an envelope is called  back of the envelope calculations. Enrico Fermi, the Italian Nobel Laureate, is associated with back of the envelope calculations as he was well known for emphasizing ways that complex scientific equations could be approximated with an order of magnitude using simple calculations.


Fermi also developed a series of calculations called ‘Fermi Questions’ to solve estimation problems known as Fermi problems. He was known for getting quick and accurate answers to problems that would seem very difficult to his peers. The most famous instance came during the first atomic bomb test in New Mexico on 16 July 1945. As the blast wave reached him, Fermi dropped bits of paper. By measuring the distance they were blown, he could compare to a previously computed table and thus estimate the bomb energy yield. He estimated 10 kilotons of TNT; the measured result was remarkably close- 18.6.

The need for Back of the envelope calculations

  • Using back-of-the envelope calculations that help put very large or very small numbers into perspective
  • Deepens  level of understanding
  • Gives practice in estimating or seeking out reasonable starting numbers.
  • Gives practice in simple quantitative tasks.

Tips for using back-of-the-envelope calculations effectively

  • Choose calculations that help in gaining a perspective on something (e.g., size, distance, rate) that is outside their normal frame of reference.
  • Pick random estimates. Ex; estimating the number of ping pong balls to fill a squash court
  • Predict an answer before calculating. This gives a sense of your own preconceptions.
  • Choose calculations that involve at least some estimation. Otherwise, it’s just an ordinary quantitative problem.
  • Find out reasonable values for items that can’t be estimated successfully.
  • After the calculations are done, explore how much difference it makes in the results to have estimated values differently.

Few examples that you can attempt. And some more.

Did You Know?

The apparent contradiction between high estimates of the probability of the existence of extraterrestrial civilization and humanity’s lack of contact with, or evidence for, such civilizations is called Fermi’s Paradox.

Question: Answer in Comments?

Radio astronomer X [in pic]lends his name to the probabilistic argument used to estimate the number of active, communicative extraterrestrial civilizations in the Milky Way galaxy. His 1961 equation summarizes the main concepts which scientists must contemplate when considering the question of other radio-communicative life. Identify him or the equation.


Bonus Read


Enrico Fermi’s famous question, now central to debates about the prevalence of extraterrestrial civilizations, arose during a luncheon conversation with Emil Konopinski, Edward Teller, and Herbert York in the summer of 1950. Fermi’s companions on that day have provided accounts of the incident.

  • ravi bhatia

    frank drake who gave the drake equation.

  • sandeep kumar

    Frank Drake

  • zoso

    Frank Drake..the Drake Equation

  • Knowqout

    Frank Drake of the Drake equation fame. He also created the Arecibo message that was broadcast in 1974 into a star cluster 21000 light years away. Carl Sagan was one of the people who helped create the message. Here is more on the Arecibo message