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Radian measure

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

Until now you measure angles in degrees. There is a second measure: radian measure (das Bogenmaß). You measure the angle as a length along the edge of a circle with radius \(1\).
📄 Open worksheet (PDF, German)

The video

In the video an arc runs along the unit circle (der Einheitskreis). You see how to convert between degrees and radians.

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

Contents of the video

  1. 0:00 Introduction: two measures
  2. 0:20 Length along the circle
  3. 0:42 Converting
  4. 1:10 Important angles
  5. 1:30 Careful, mistake!
  6. 1:54 Try it yourself

The most important points

Length along the circle

The circumference of the unit circle is \(2\pi \approx 6{,}28\). Full circle: \(360^\circ = 2\pi\). Quarter circle: \(90^\circ = \frac{\pi}{2}\). Half circle: \(180^\circ = \pi\).

Arc lengths on the unit circle: \pi/2 and \pi.
Arc lengths on the unit circle: \(\frac{\pi}{2}\) and \(\pi\).

Converting

\(180^\circ\) correspond to \(\pi\). Degrees \(\to\) radians: \(x = \alpha \cdot \frac{\pi}{180^\circ}\). Radians \(\to\) degrees: \(\alpha = x \cdot \frac{180^\circ}{\pi}\). Example: \(60^\circ \cdot \frac{\pi}{180^\circ} = \frac{\pi}{3}\).

Converting degrees to radians.
Converting degrees to radians.

Important angles

\(30^\circ = \frac{\pi}{6}\), \(45^\circ = \frac{\pi}{4}\), \(60^\circ = \frac{\pi}{3}\), \(90^\circ = \frac{\pi}{2}\), \(180^\circ = \pi\), \(270^\circ = \frac{3\pi}{2}\), \(360^\circ = 2\pi\).

The important angles in degrees and in radians.
The important angles in degrees and in radians.

Careful: Not \(\pi = 180\)! It is true that \(\pi \approx 3{,}14\), but the angle \(\pi\) is \(180^\circ\). In GeoGebra sin(90°) calculates in degrees, sin(90) in radians.

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

Convert \(120^\circ\) to radians. Then convert \(\frac{3\pi}{4}\) to degrees.

Show solution

\(120^\circ \cdot \frac{\pi}{180^\circ} = \frac{2\pi}{3}\). \(\frac{3\pi}{4} \cdot \frac{180^\circ}{\pi} = 135^\circ\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: das Bogenmaß (Grad und Bogenmaß umrechnen, Einheitskreis). 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.