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The Pythagorean theorem

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

The most famous theorem in school maths: \( a^2 + b^2 = c^2 \). With squares on the sides, a knotted rope, a proof with two big squares – and it is older than Pythagoras.
📄 Open worksheet (PDF, German)

The video

Watch the colours: \( a \) is blue, \( b \) is purple, the hypotenuse \( c \) is yellow, the triangles in the proof are orange.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:20 The right-angled triangle
  3. 0:44 Squares on the sides
  4. 1:11 The Pythagorean theorem
  5. 1:28 Older than Pythagoras
  6. 1:53 The knotted rope
  7. 2:28 The proof
  8. 3:33 Calculating the hypotenuse
  9. 4:06 Careful, mistake!
  10. 4:25 Try it yourself

The most important points

In a right-angled triangle the longest side is the hypotenuse (die Hypotenuse) \( c \). It is opposite the right angle. The two other sides are the legs (die Katheten) \( a \) and \( b \).

Squares on the sides: 9 + 16 = 25 .
Squares on the sides: \( 9 + 16 = 25 \).

The Pythagorean theorem (der Satz des Pythagoras)

In every right-angled triangle \( a^2 + b^2 = c^2 \). The squares on the legs together are as big as the square on the hypotenuse. Example: \( 6^2 + 8^2 = 100 \), so \( c = \sqrt{100} = 10 \).

The theorem is older than Pythagoras (about 500 BC): already in Babylon (about 1800 BC) a clay tablet shows numbers that fit it exactly. Whether Pythagoras proved it, nobody knows.

Knotted rope with 12 sections: 3^2 + 4^2 = 5^2 gives a right angle. That the rope stretchers in ancient Egypt worked like this is only a guess.
Knotted rope with 12 sections: \( 3^2 + 4^2 = 5^2 \) gives a right angle. That the rope stretchers in ancient Egypt worked like this is only a guess.
Proof: left 4 triangles + c^2 , right 4 triangles + a^2 + b^2 . Without the triangles, c^2 = a^2 + b^2 is left.
Proof: left 4 triangles + \( c^2 \), right 4 triangles + \( a^2 + b^2 \). Without the triangles, \( c^2 = a^2 + b^2 \) is left.

Careful: the theorem only works in a right-angled triangle, and \( c \) is always the longest side. From \( c^2 = 100 \) it follows that \( c = 10 \), not 50.

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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Show the worksheet solutions
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Try it yourself

Task

Calculate the hypotenuse: a) \( a = 3,\ b = 4 \) b) \( a = 5,\ b = 12 \) c) \( a = 8,\ b = 15 \)

Show solution

a) \( 5 \) b) \( 13 \) c) \( 17 \)

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: the Pythagorean theorem (a² + b² = c², hypotenuse and legs). Give me five short tasks, one after the other. Always wait for my answer. Then tell me if it is right and explain briefly why. Write in simple German and put an English translation in brackets after each sentence.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.