Cones, Cylinders, Spheres

Archimedes’ favourite result

An interactive lesson with 8 steps, about 7 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• Cylinder $\pi r^2 h$. Cone (and pyramid) $\tfrac13$ of the matching prism.
• Cavalieri: equal slices at every height mean equal volumes.
• Sphere: $\tfrac43\pi r^3$, two-thirds of its surrounding cylinder (Archimedes).

Steps

1. Cylinder
2. A cone is a third
3. Cavalieri’s principle
4. The sphere
5. Archimedes’ slices
6. Ball volume
7. Archimedes’ favourite
8. Scaling up