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Back-of-the-envelope calculation

The phrase back-of-the-envelope calculations refers to rough calculations that, while not rigorous, test or support a point. They are far more than a guess but far less than a proof. The phrase is generally used in mathematics, physics and engineering. It refers to the practice of quickly jotting down calculations on the nearest available piece of paper, such as the back of an envelope.

The term is an idiom; back-of-the-envelope calculations need not be performed on an envelope, or written on paper, or even written down at all. The defining factor is the use of simplified, scaled-down models.


In the hard sciences, back-of-the-envelope calculation is often associated with physicist Enrico Fermi, who was well known for emphasizing ways that complex scientific equations could be approximated within an order of magnitude using simple calculations. He went on to develop a series of sample calculations which are called "Fermi Questions" or "Back-of-the-Envelope Calculations" and used to solve Fermi problems.

Fermi was known for getting quick and accurate answers to problems which would stump other people. The most famous instance came during the first atomic bomb test in New Mexico on July 16 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 18.6.

Another example is Victor Weisskopf's pamphlet Modern Physics from an Elementary Point of View. In these notes Weisskof used back-of-the-envelope calculations to calculate the size of a hydrogen atom, a star, and a mountain, all using elementary physics.

A similar phrase is back of a napkin.


  • When British engineer James Nasmyth was asked to build a bigger hammer after the workpiece was too big for existing fall hammers, Nasmyth, after a little thought, sketched out on a piece of paper the design for the steam hammer.
  • There is a commonly held legend that economist Arthur Laffer first explained the Laffer Curve on the back of a napkin.
  • Legend has it that the original Southwest Airlines "Texas Triangle", connecting its major cities (Dallas, Houston, San Antonio) was created on the back of a napkin.

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