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Arc and sector

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

A piece of cake has the shape of a sector (der Kreissektor). Arc (der Kreisbogen) and area are a fraction of the whole circle: \(\frac{\alpha}{360^\circ}\).
📄 Open worksheet (PDF, German)

The video

In the video we calculate arc and area for \(\alpha = 90^\circ\) and \(\alpha = 60^\circ\).

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 arc
  3. 0:47 The sector
  4. 1:09 An example
  5. 1:35 Careful, mistake!
  6. 1:55 Try it yourself

The most important points

Arc

\(b = \frac{\alpha}{360^\circ} \cdot 2\pi r\). With \(r = 4\) cm and \(\alpha = 90^\circ\): \(b = \frac{1}{4} \cdot 2 \cdot \pi \cdot 4 \approx 6{,}28\) cm.

The arc is a quarter of the circumference.
The arc is a quarter of the circumference.

Sector

\(A = \frac{\alpha}{360^\circ} \cdot \pi r^2\). With \(r = 4\) cm: \(A = \frac{1}{4} \cdot \pi \cdot 16 \approx 12{,}57\,\text{cm}^2\).

The area is a quarter of the circle's area.
The area is a quarter of the circle's area.

\(\alpha = 60^\circ\), \(r = 3\) cm: fraction \(\frac{60}{360} = \frac{1}{6}\), \(b = \pi \approx 3{,}14\) cm, \(A \approx 4{,}71\,\text{cm}^2\).

One sixth of the circle.
One sixth of the circle.

Careful: not \(\frac{60}{100}\)! A circle has \(360^\circ\).

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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Show the worksheet solutions
The worksheet – Page 4
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Try it yourself

Task

A sector has \(r = 6\) cm and \(\alpha = 120^\circ\). Calculate the arc \(b\) and the area \(A\).

Show solution

\(\frac{120}{360} = \frac{1}{3}\): \(b = 4\pi \approx 12{,}57\) cm, \(A = 12\pi \approx 37{,}70\,\text{cm}^2\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Kreisbogen und Kreissektor (Anteil α/360°, b = α/360° · 2πr, A = α/360° · πr²). 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.