ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Pythagoras' Theorem - A Proof

Updated on June 11, 2019
David3142 profile image

I am a former maths teacher and owner of Doingmaths. I love writing about maths, its applications and fun mathematical facts.

A right-angled triangle

What is Pythagoras' Theorem?

Pythagoras' Theorem states that for any right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Put algebraically, using our diagram above where a and b are our two perpendicular sides and c is the hypotenuse, we get a2 + b2 = c2.

Pythagoras' Theorem has many important applications across mathematics from simple geometry through to trigonometry and can even be used with n-dimensional solids!

It was known by the ancient Babylonians and Egyptians as far back as 1900 BC and Pythagorean triples (whole numbers that satisfy the equation such as 32 + 42 = 52 and 52 + 122 = 132) can be found on the Plimpton 322 tablet, a Babylonian clay tablet dating from approximately 1800 BC. It is also believed that ancient Egpytian builders may have used rope with equally spaced knots and the knowledge that a triangle with sides of 3, 4 and 5 is right-angled to ensure the accuracy of right-angles when constructing the pyramids.

Pythagoras' theorem was also discovered independently in other cultures around the world such as Mesoptamia, China and India, however it is the ancient Greeks who are most commonly associated with this theorem.

A bust of Pythagoras in the Vatican Museum, Rome.

Pythagoras of Samos

The theorem is named after the ancient Greek mathematician and philosopher Pythagoras (c 569 - 495 BC). Although earlier civilizations where aware of aspects of the theorem, it is Pythagoras who is credited with the first proof of the theorem, although no evidence of this proof remains. Interestingly, as the Greeks were much more adept at geometry than they were at algebra, Pythagoras would not have thought of the theorem as an algebraic one, but instead as a triangle with a square attached to each side, where the areas of the two smaller squares added up to the area of the larger square on the hypotenuse.

Little reliable evidence about Pythagoras' life exists today and when researching him, it is very difficult to separate fact from fiction. Many mathematical and scientific discoveries are attributed to him such as irrational numbers, the regular solids, and of course the theorem that bears his name, but again, it is difficult to know what was actually Pythagoras and what was discovered by his followers and pupils.

Proving Pythagoras' Theorem

There are many ways to prove Pythagoras' Theorem and in this article we are going to use a quick, concise one which uses geometry and some simple algebra.

To begin with look at the diagram below which consists of a large square with a smaller square inside it, angled to create four right-angled triangles around its edges.

Two squares, one inside the other.

Pythagoras' Theorem proof continued

If we label one of the triangles so that the perpendicular sides are a and b, and the hypotenuse is c, we can quickly see that the remaining lengths in the diagram must also all be a, b and c.

This can be demonstrated by using the fact that angles in a triangle and angles on a straight line both add up to 180°. Using these facts we can quickly see that the angles in the triangles are all the same, hence the four triangles are similar. Furthermore, as each triangle has a side of the smaller square as a hypotenuse, the hypotenuses are all the same length, hence the four triangles must all be congruent (identical in size).

Using this fact, we can now label all of the lengths with a, b or c.

Square diagram with labelled edges

Pythagoras' Theorem Proof - The Algebra bit

We are now going to calculate the area of the large square (you will see why soon).

Each side is of length a + b, so we get:

Area of large square = (a + b)2

= a2 + b2 + 2ab

We can also express the area of the large square as the areas of the small square (c2) and the four triangles (1/2 × ab).

Area of large square = c2 + 4 ×1/2 × ab

= c2 + 2ab

As these two expressions are both the area of the large square they must be equal to each other so:

a2 + b2 + 2ab = c2 + 2ab

and cancelling 2ab from each side leaves us with:

a2 + b2 = c2

which is Pythagoras' theorem. Proof complete.

A video version of this article from the DoingMaths channel on YouTube

© 2019 David


    0 of 8192 characters used
    Post Comment
    • David3142 profile imageAUTHOR


      5 months ago from West Midlands, England

      Thank you. I particularly like how the ancient Egyptians used Pythagoras' theorem and knotted rope to make right-angled triangles which they could then use to measure accurate right angles in construction.

    • verdict profile image

      George Dimitriadis 

      5 months ago from Templestowe


      A good summary of Pythagoras' Theorem. Perhaps you might consider including one or two practical applications of the Theorem.


    • EstX Neyo profile image

      Mikey Karlovsky 

      21 months ago

      Looking back at it, (now going into Honors Pre Calc) I wish this article was present when I was in Geometry

    • David3142 profile imageAUTHOR


      22 months ago from West Midlands, England

      Thanks :-)

    • Larry Slawson profile image

      Larry Slawson 

      22 months ago from North Carolina

      Very cool. Thank you for sharing.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
    ClickscoThis is a data management platform studying reader behavior (Privacy Policy)