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Surface area of the pyramid

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

You unfold a pyramid into a net: a square and four triangles. For the triangles you need the height of the side face.
📄 Open worksheet (PDF, German)

The video

In the video a pyramid is unfolded and folded again.

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 net
  3. 0:45 The height of the side face
  4. 1:08 The surface area
  5. 1:48 With the Pythagorean theorem
  6. 2:16 Careful, mistake!
  7. 2:35 Try it yourself

The most important points

The net of a square pyramid: a square (the base \(G\)) and four equal triangles (the lateral surface \(M\), der Mantel).

The net of the pyramid.
The net of the pyramid.

Two heights

The height of the pyramid \(h\) goes straight up inside. For the triangles you need the height of the side face \(h_s\) (die Höhe der Seitenfläche). Pythagoras: \(h_s^2 = h^2 + (a : 2)^2\).

h = 2 cm inside, h_s = 2{,}5 cm on the triangle.
\(h = 2\) cm inside, \(h_s = 2{,}5\) cm on the triangle.

The surface area

\(O = G + M\). With \(a = 3\) cm, \(h_s = 2{,}5\) cm: \(G = 9\text{ cm}^2\), \(M = 4 \cdot 3 \cdot 2{,}5 : 2 = 15\text{ cm}^2\), \(O = 24\text{ cm}^2\).

The faces in the net.
The faces in the net.

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

Square pyramid: \(a = 6\) cm, \(h_s = 5\) cm. Calculate the surface area.

Show solution

\(O = 36 + 4 \cdot 15 = 96\text{ cm}^2\)

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Oberfläche der Pyramide (Netz, Grundfläche, Mantel, Höhe der Seitenfläche, O = G + M). 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.