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Surface area of the cone

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

The lateral surface of a cone is a sector of a circle. Cut into pieces it becomes almost a rectangle – this is how you find the formula.
📄 Open worksheet (PDF, German)

The video

In the video a cone rolls around its apex. Then the lateral surface is cut into pieces and rearranged.

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 slant height
  3. 0:53 The net
  4. 1:27 The lateral surface
  5. 2:09 The surface area
  6. 2:30 Careful, mistake!
  7. 2:48 Try it yourself

The most important points

The slant height

The slant height \(s\) (die Mantellinie) goes from the apex to the edge. Pythagoras: \(s^2 = r^2 + h^2\). Example: \(r = 3\) cm, \(h = 4\) cm, \(s = 5\) cm.

h, r and s form a right triangle.
\(h\), \(r\) and \(s\) form a right triangle.

The net: a circle and a sector (der Kreisausschnitt) with radius \(s\). The arc is as long as the circumference \(2 \cdot \pi \cdot r\).

The net of the cone.
The net of the cone.

Lateral surface and surface area

\(M = \pi \cdot r \cdot s\), \(O = \pi \cdot r^2 + \pi \cdot r \cdot s\). Here: \(M \approx 47{,}1\text{ cm}^2\), \(O \approx 75{,}4\text{ cm}^2\).

Almost a rectangle: \pi · r long, s high.
Almost a rectangle: \(\pi \cdot r\) long, \(s\) high.

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
Page 1/3
The worksheet – Page 2
Page 2/3
Show the worksheet solutions
The worksheet – Page 3
Page 3/3

Try it yourself

Task

Use \(\pi \approx 3{,}14\). Cone: \(r = 6\) cm, \(s = 10\) cm. Calculate the surface area.

Show solution

\(O \approx 113{,}04 + 188{,}4 = 301{,}44\text{ cm}^2\)

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Oberfläche des Kegels (Mantellinie s, Kreisausschnitt, M = π · r · s, O = π · r² + π · r · s). 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.