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Circumference and area: circle

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

You get to know the number \(\pi\). Then the circle is cut into more and more pieces – until it is almost a rectangle.
📄 Open worksheet (PDF, German)

The video

In the video a circle rolls along a line. Then 8, 16 and 32 pieces of the circle are laid side by side.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:16 The circumference
  3. 0:38 The number π
  4. 1:12 Cutting and laying
  5. 1:29 More and more pieces
  6. 1:56 The formula
  7. 2:32 Careful, mistake!
  8. 2:58 Try it yourself

The most important points

The circumference

For every circle the circumference (der Umfang) is about 3.14 times as long as the diameter. This number is called \(\pi\) (pi), \(\pi \approx 3{,}14\). Circumference: \(u = \pi \cdot d = 2 \cdot \pi \cdot r\).

A circle with d = 3 cm rolls one turn: u \approx 9{,}4 cm.
A circle with \(d = 3\) cm rolls one turn: \(u \approx 9{,}4\) cm.

We cut the circle into pieces and lay them side by side, one up, one down. The more pieces, the more the shape becomes a rectangle.

32 pieces: almost a rectangle.
32 pieces: almost a rectangle.

Area

Height: \(r\). Length: half the circumference \(\pi \cdot r\). \(A = \pi \cdot r \cdot r = \pi \cdot r^2\). Careful: the radius goes into the formula!

r = 3 cm: A \approx 3{,}14 · 9 \approx 28{,}3\text{ cm}^2.
\(r = 3\) cm: \(A \approx 3{,}14 \cdot 9 \approx 28{,}3\text{ cm}^2\).

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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The worksheet – Page 2
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The worksheet – Page 3
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The worksheet – Page 4
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Show the worksheet solutions
The worksheet – Page 5
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Try it yourself

Task

Use \(\pi \approx 3{,}14\). a) circumference for \(d = 10\) cm b) area for \(r = 10\) cm c) area for \(d = 8\) cm

Show solution

a) \(31{,}4\) cm b) \(314\text{ cm}^2\) c) \(r = 4\) cm, \(A \approx 50{,}2\text{ cm}^2\)

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Umfang und Flächeninhalt des Kreises (Radius, Durchmesser, Pi, u = π · d, A = π · r²). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

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.