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Volume of the sphere

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

Why does a sphere fill two thirds of the cylinder? With Cavalieri's principle and a trick by Archimedes you find the formula.
📄 Open worksheet (PDF, German)

The video

In the video a hemisphere and a cylinder without a cone are cut at every height.

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 Archimedes
  3. 0:44 Cavalieri's principle
  4. 1:28 Two solids
  5. 1:50 The cross-section
  6. 3:11 The formula
  7. 3:37 Sphere and cylinder
  8. 4:31 Try it yourself

The most important points

Cavalieri's principle

If two solids have equal cross-sections at every height, they have the same volume. Example: a straight and a slanted stack of equal coins.

Two stacks of coins with the same volume.
Two stacks of coins with the same volume.

Hemisphere and cylinder without cone

Cut at height \(h\): on the left a circle \(\pi \cdot (r^2 - h^2)\), on the right a ring \(\pi \cdot r^2 - \pi \cdot h^2\). Equal! So: \(V_\text{hemisphere} = \pi r^3 - \frac{1}{3}\pi r^3 = \frac{2}{3}\pi r^3\) and \(V_\text{sphere} = \frac{4}{3} \cdot \pi \cdot r^3\).

Equal cross-sections at every height.
Equal cross-sections at every height.

Archimedes (about 287–212 BC, Syracuse) showed: sphere and cylinder have the ratio \(2 : 3\). According to Plutarch he wanted a sphere and a cylinder on his tomb; Cicero reports that he found the tomb in 75 BC.

Sphere : cylinder = 2 : 3.
Sphere : cylinder = 2 : 3.

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

Use \(\pi \approx 3{,}14\). What is the volume of a sphere with a) \(r = 3\) cm b) \(r = 10\) cm?

Show solution

a) \(113{,}04\text{ cm}^3\) b) \(\approx 4186{,}7\text{ cm}^3\)

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Volumen der Kugel (Prinzip von Cavalieri, Halbkugel = Zylinder minus Kegel, V = 4/3 · π · r³, Archimedes). 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.