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The incircle

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

You learn the incircle (der Inkreis): it lies inside the triangle and touches all three sides. Its centre lies on the angle bisectors.
📄 Open worksheet (PDF, German)

The video

In the video the three angle bisectors meet in the centre \(I\). The perpendicular to one side is the radius.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:09 The angle bisectors
  3. 0:26 The radius
  4. 0:50 The description
  5. 1:10 Careful, mistake!
  6. 1:30 Try it yourself

The most important points

The incircle

The incircle touches all three sides from inside. Its centre \(I\) is the intersection of the three angle bisectors.

The triangle with its incircle.
The triangle with its incircle.

The radius

\(I\) has the same distance from all three sides. This distance (the perpendicular from \(I\) to a side) is the radius \(r\).

The three equal distances are the radius.
The three equal distances are the radius.

Construction

1. Construct two angle bisectors: intersection \(I\). 2. Drop the perpendicular from \(I\) to a side. 3. Draw the circle around \(I\) through the foot point.

Careful: Do not mix up circumcircle and incircle. The circumcircle goes through the corners (centre on the perpendicular bisectors). The incircle touches the sides (centre on the angle bisectors).

Circumcircle and incircle in the same triangle.
Circumcircle and incircle in the same triangle.

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 right triangle has the sides \(3\) cm, \(4\) cm and \(5\) cm. How big is the radius of the incircle? Hint: \(r = \frac{2 \cdot A}{u}\).

Show solution

\(A = \frac{3 \cdot 4}{2} = 6\) cm², \(u = 12\) cm, \(r = \frac{2 \cdot 6}{12} = 1\) cm.

Keep learning with a language model

You are my maths teacher. I am learning German as a second language. Our topic: Inkreis (Winkelhalbierende, Radius, Konstruktion). 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. Use simple German sentences.

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.