Back-of-the-envelope calculation

<h2 id="history">History</h2>
In the natural sciences, back-of-the-envelope calculation is often associated with physicist <a href="/facts/Enrico_Fermi/cmYcY15H">Enrico Fermi</a>,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> who was well known for emphasizing ways that complex scientific equations could be approximated within an <a href="/facts/Order_of_magnitude/cWzRP2dM">order of magnitude</a> 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 <a href="/facts/Fermi_problem/LueugRQk">Fermi problems</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
Fermi was known for getting quick and accurate answers to problems that would stump other people. The most famous instance came during the <a href="/facts/First_atomic_bomb/0aHtpyUN">first atomic bomb</a> test in <a href="/facts/New_Mexico/wosJCGaW">New Mexico</a> 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 18.6.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
Perhaps the most influential example of such a calculation was carried out over a period of a few hours by <a href="/facts/Arnold_Wilkins/Djj43gcY">Arnold Wilkins</a> after being asked to consider a problem by <a href="/facts/Robert_Watson_Watt/uxtmCwH9">Robert Watson Watt</a>. Watt had learned that the Germans claimed to have invented a radio-based death ray, but Wilkins' one-page calculations demonstrated that such a thing was almost certainly impossible. When Watt asked what role radio might play, Wilkins replied that it might be useful for detection at long range, a suggestion that led to the rapid development of <a href="/facts/Radar/jk9BTbFA">radar</a> and the <a href="/facts/Chain_Home/XPwgbaQr">Chain Home</a> system.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
Another example is <a href="/facts/Victor_Weisskopf/jocTeNnB">Victor Weisskopf</a>'s pamphlet Modern Physics from an Elementary Point of View.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> In these notes Weisskopf used back-of-the-envelope calculations to calculate the size of a hydrogen atom, a star, and a mountain, all using elementary physics.

<h2 id="examples">Examples</h2>
In a video interview for the <a href="/facts/University_of_California%2c_Berkeley/MXVEjklr">University of California, Berkeley</a> on the 50th anniversary of the laser, Nobel laureate <a href="/facts/Charles_Townes/sRWs7U4e">Charles Townes</a> described how he pulled an envelope from his pocket while sitting in a park and wrote down calculations during his initial insight into lasers.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>
During lunch with NFL commissioner <a href="/facts/Pete_Rozelle/ANXtc6d0">Pete Rozelle</a> in 1966, <a href="/facts/Tiffany_%2526_Co./yKNRwewI">Tiffany & Co.</a> vice president Oscar Riedner made a sketch on a cocktail napkin of what would become the <a href="/facts/Vince_Lombardi_Trophy/HHAxMzGS">Vince Lombardi Trophy</a>, awarded annually to the winner of the <a href="/facts/Super_Bowl/W2dqzqY4">Super Bowl</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>
An important Internet protocol, the <a href="/facts/Border_Gateway_Protocol/XfYgBzW3">Border Gateway Protocol</a>, was sketched out in 1989 by engineers on the back of "three ketchup-stained napkins", and is still known as the three-napkin protocol.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
<a href="/facts/UTF-8/xlaVHSyf">UTF-8</a>, the dominant character encoding for the <a href="/facts/World_Wide_Web/wvnTrWFq">World Wide Web</a>,<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> was designed by <a href="/facts/Ken_Thompson/KQtrw9YM">Ken Thompson</a> and <a href="/facts/Rob_Pike/K5R9UoSH">Rob Pike</a> on a placemat.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>
The <a href="/facts/Bailey_bridge/w9YY9Abv">Bailey bridge</a> is a type of portable, pre-fabricated, truss bridge and was extensively used by British, Canadian and US military engineering units. <a href="/facts/Donald_Bailey_(civil_engineer)/mqFTpLJG">Donald Bailey</a> drew the original design for the bridge on the back of an envelope.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>
The <a href="/facts/Laffer_Curve/WBYRLXHy">Laffer Curve</a>, which claims to show the relationship between tax cuts and government income, was drawn by <a href="/facts/Arthur_Laffer/5cMHTGdk">Arthur Laffer</a> in 1974 on a bar napkin to show an aide to President <a href="/facts/Gerald_R._Ford/AdMS7jGW">Gerald R. Ford</a> why the federal government should cut taxes.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>
Upon hearing that the <a href="/facts/Saturn_I/5Tp3eQ72">S-IV 2nd Stage</a> of the <a href="/facts/Saturn_I/5Tp3eQ72">Saturn I</a> would need transport from California to Florida for launch as part of the <a href="/facts/Apollo_program/IC1Ek1Ao">Apollo program</a>, <a href="/facts/John_M._Conroy/yCVlwhed">Jack Conroy</a> sketched the cavernous cargo airplane, the <a href="/facts/Aero_Spacelines_Pregnant_Guppy/muTma1Kz">Pregnant Guppy</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>
The <a href="/facts/Video_Toaster/JBltVpk2">Video Toaster</a> was designed on placemats in a Topeka pizza restaurant.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Buckingham_pi_theorem/R3j8Dl2w">Buckingham pi theorem</a>, a technique often used in <a href="/facts/Fluid_mechanics/NIy4tRfE">fluid mechanics</a> to obtain order-of-magnitude estimates</li>
<li><a href="/facts/Guesstimate/1mVeqQSm">Guesstimate</a></li>
<li><a href="/facts/Scientific_Wild-Ass_Guess/h5Hqk5gj">Scientific Wild-Ass Guess</a></li>
<li><a href="/facts/Heuristic/BcBEkv9c">Heuristic</a></li>
<li><a href="/facts/Order-of-magnitude_analysis/xwq3Mxej">Order-of-magnitude analysis</a></li>
<li><a href="/facts/Rule_of_thumb/RopfGsJ0">Rule of thumb</a></li>
<li><a href="/facts/Sanity_testing/4l0uVCdR">Sanity testing</a></li>
<li><a href="/facts/Fermi_Problem/LueugRQk">Fermi Problem</a></li></ul>
<h2 id="notes-and-references">Notes and references</h2>

<h2 id="external-links">External links</h2>

Look up back-of-the-envelope or back-of-an-envelope in Wiktionary, the free dictionary.

<ul><li><a href="http://www-cse.ucsd.edu/classes/sp97/cse141/botec.html">Syllabus at UCSD</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Brown, Bob (2011-07-19). "Napkins: Where Ethernet, Compaq and Facebook's cool data center got their starts". Network World. Archived from the original on 15 August 2019. Retrieved 2020-10-06. Robert Metcalfe's early Ethernet diagrams from his days at Xerox PARC back in the early 1970s might be the most famous napkin sketches in the technology industry. <a href="https://www.networkworld.com/article/2220218/napkins--where-ethernet--compaq-and-facebook-s-cool-data-center-got-their-starts.html" target="_blank">https://www.networkworld.com/article/2220218/napkins--where-ethernet--compaq-and-facebook-s-cool-data-center-got-their-starts.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Where Fermi stood. - Bulletin of the Atomic Scientists | Encyclopedia.com (Archived) <a href="https://web.archive.org/web/20080219101037/http://www.encyclopedia.com/doc/1G1-78334537.html" target="_blank">https://web.archive.org/web/20080219101037/http://www.encyclopedia.com/doc/1G1-78334537.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Back of the Envelope Calculations <a href="http://serc.carleton.edu/quantskills/teaching_methods/boe/index.html" target="_blank">http://serc.carleton.edu/quantskills/teaching_methods/boe/index.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">High School Mathematics at Work: Essays and Examples for the Education of All Students <a href="http://www.nap.edu/html/hs_math/be.html" target="_blank">http://www.nap.edu/html/hs_math/be.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Rhodes, Richard (1986). The Making of the Atomic Bomb. New York: Simon & Schuster. p. 674. ISBN 978-0-671-44133-3. OCLC 13793436. <a href="978-0-671-44133-3" target="_blank">978-0-671-44133-3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">"Nuclear Weapons Journal, Los Alamos National Laboratory, Issue 2 2005" (PDF). Archived from the original (PDF) on 2018-12-29. Retrieved 2014-09-07. <a href="https://web.archive.org/web/20181229223636/https://www.lanl.gov/science/weapons_journal/wj_pubs/11nwj2-05.pdf" target="_blank">https://web.archive.org/web/20181229223636/https://www.lanl.gov/science/weapons_journal/wj_pubs/11nwj2-05.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Austin, B.A. (1999). "Precursors To Radar — The Watson-Watt Memorandum And The Daventry Experiment" (PDF). International Journal of Electrical Engineering & Education. 36 (4): 365–372. doi:10.7227/IJEEE.36.4.10. S2CID 111153288. Archived from the original (PDF) on 2015-05-25. Retrieved 2016-07-08. <a href="https://web.archive.org/web/20150525040134/http://www.bawdseyradar.org.uk/wp-content/uploads/2012/12/Wilkins-Calculations.pdf" target="_blank">https://web.archive.org/web/20150525040134/http://www.bawdseyradar.org.uk/wp-content/uploads/2012/12/Wilkins-Calculations.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Lectures given in the 1969 Summer Lecture Programme, CERN (European Organization for Nuclear Research), CERN 70-8, 17 March 1970. <a href="http://cdsweb.cern.ch/record/274976/" target="_blank">http://cdsweb.cern.ch/record/274976/</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Video of interview with Charles Townes; envelope mention comes about halfway in <a href="http://laserfest.org/lasers/video-history.cfm" target="_blank">http://laserfest.org/lasers/video-history.cfm</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">"Vince Lombardi Trophy". ProFootballHOF.com. NFL Enterprises, LLC. Retrieved November 21, 2019. <a href="https://www.profootballhof.com/news/vince-lombardi-trophy/" target="_blank">https://www.profootballhof.com/news/vince-lombardi-trophy/</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Timberg, Craig (31 May 2015). "Net of Insecurity; Quick fix for an early Internet problem lives on a quarter-century later". The Washington Post. Archived from the original on 1 June 2015. Retrieved 4 January 2021. As the prospect of system meltdown loomed, the men began scribbling ideas for a solution onto the back of a ketchup-stained napkin. Then a second. Then a third. The "three-napkins protocol," as its inventors jokingly dubbed it, would soon revolutionize the Internet. And though there were lingering issues, the engineers saw their creation as a "hack" or "kludge," slang for a short-term fix to be replaced as soon as a better alternative arrived. <a href="http://www.washingtonpost.com/sf/business/2015/05/31/net-of-insecurity-part-2/" target="_blank">http://www.washingtonpost.com/sf/business/2015/05/31/net-of-insecurity-part-2/</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">"Usage Survey of Character Encodings broken down by Ranking". w3techs.com. Retrieved 2018-11-01. <a href="https://w3techs.com/technologies/cross/character_encoding/ranking" target="_blank">https://w3techs.com/technologies/cross/character_encoding/ranking</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Email Subject: UTF-8 history, From: "Rob 'Commander' Pike", Date: Wed, 30 Apr 2003..., ...UTF-8 was designed, in front of my eyes, on a placemat in a New Jersey diner one night in September or so 1992...So that night Ken wrote packing and unpacking code and I started tearing into the C and graphics libraries. The next day all the code was done... <a href="https://www.cl.cam.ac.uk/~mgk25/ucs/utf-8-history.txt" target="_blank">https://www.cl.cam.ac.uk/~mgk25/ucs/utf-8-history.txt</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Services, Times Wire (1985-05-07). "Sir Donald Bailey, WW II Engineer, Dies". Los Angeles Times. ISSN 0458-3035. Retrieved 2018-11-01. He sketched the original design for the Bailey Bridge on the back of an envelope as he was being driven to a meeting of Royal Engineers to debate the failure of existing portable bridges <a href="https://www.latimes.com/archives/la-xpm-1985-05-07-mn-11177-story.html" target="_blank">https://www.latimes.com/archives/la-xpm-1985-05-07-mn-11177-story.html</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">"This is not Arthur Laffer's famous napin" NY Times 13 Oct. 2017 <a href="https://www.nytimes.com/2017/10/13/us/politics/arthur-laffer-napkin-tax-curve.html?module=WatchingPortal®ion=c-column-middle-span-region&pgType=Homepage&action=click&mediaId=thumb_square&state=standard&contentPlacement=2&version=internal&contentCollection=www.nytimes.com&contentId=https%3A%2F%2Fwww.nytimes.com%2F2017%2F10%2F13%2Fus%2Fpolitics%2Farthur-laffer-napkin-tax-curve.html&eventName=Watching-article-click&_r=0" target="_blank">https://www.nytimes.com/2017/10/13/us/politics/arthur-laffer-napkin-tax-curve.html?module=WatchingPortal®ion=c-column-middle-span-region&pgType=Homepage&action=click&mediaId=thumb_square&state=standard&contentPlacement=2&version=internal&contentCollection=www.nytimes.com&contentId=https%3A%2F%2Fwww.nytimes.com%2F2017%2F10%2F13%2Fus%2Fpolitics%2Farthur-laffer-napkin-tax-curve.html&eventName=Watching-article-click&_r=0</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Bloom, Margy (15 September 2011). "PilotMag Aviation Magazine | The Pregnant Guppy | The Problem: Logistics". Pilot Magazine. Archived from the original on 15 July 2011. Retrieved 25 April 2012. [Conroy] listened to the conversations around him, then picked up a cocktail napkin and a ballpoint pen. And with the precision he'd learned during the brief months he'd attended engineering school many years before, he drew an airplane that had never been built, to carry a rocket that had never been launched, to take man to a place nobody had ever been before. Jack Conroy had just sketched the airplane that would become the Pregnant Guppy. <a href="https://web.archive.org/web/20110715083759/http://pilotmag.com/guppy" target="_blank">https://web.archive.org/web/20110715083759/http://pilotmag.com/guppy</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Reimer, Jeremy (18 March 2016). "A history of the Amiga, part 9: The Video Toaster". Ars Technica. Archived from the original on 18 March 2016. Retrieved 4 January 2021. Montgomery suggested that Jenison meet his friend Brad Carvey, who had been working on projects involving robotic vision. The three of them got together in a pizza restaurant in Topeka and started drawing block diagrams on the placemats. <a href="https://arstechnica.com/gadgets/2016/03/a-history-of-the-amiga-part-9-the-video-toaster/" target="_blank">https://arstechnica.com/gadgets/2016/03/a-history-of-the-amiga-part-9-the-video-toaster/</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
</ol>

Back-of-the-envelope calculation open-in-new

Back-of-the-envelope calculation