targas-edu.de
DE ☾ A+

Surface area of the sphere

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

A sphere has no net. With very many small pyramids you still find its surface area.
📄 Open worksheet (PDF, German)

The video

In the video the sphere is divided into many small pyramids. Their apexes are at the centre.

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

Contents of the video

  1. 0:00 Introduction
  2. 0:19 Many small pyramids
  3. 1:07 The calculation
  4. 2:20 Four circles
  5. 2:37 Sphere and cylinder
  6. 3:05 Careful, mistake!
  7. 3:29 Try it yourself

The most important points

We divide the sphere into very many small pyramids: apex at the centre, base on the surface, height \(r\).

The more pyramids, the more exact.
The more pyramids, the more exact.

The calculation

\(V_\text{sphere} = \frac{1}{3} \cdot r \cdot (A_1 + \dots + A_n) = \frac{1}{3} \cdot r \cdot O\). With \(V = \frac{4}{3} \cdot \pi \cdot r^3\): \(\frac{1}{3} \cdot r \cdot O = \frac{4}{3} \cdot \pi \cdot r^3\), so \(O = 4 \cdot \pi \cdot r^2\).

Rearranging: · 3, then : r.
Rearranging: \(\cdot 3\), then \(: r\).

The surface area is as big as four circles with radius \(r\).

O = 4 · \pi · r^2.
\(O = 4 \cdot \pi \cdot r^2\).

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 \(\pi \approx 3{,}14\). What is the surface area of a sphere with a) \(r = 5\) cm b) \(d = 6\) cm?

Show solution

a) \(314\text{ cm}^2\) b) \(113{,}04\text{ cm}^2\)

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Oberfläche der Kugel (O = 4 · π · r², Herleitung mit vielen kleinen Pyramiden). 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.