Round all the way
A steel bowl is half of a ball, cut straight through. Slide the cut down. The flat face grows until the cut reaches the centre.
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- A sphere, radius r.
- Cut it straight through the centre.
- Each half is a hemisphere, with one flat face.
- The flat face is new. The sphere had none.
The rule
A sphere is a round solid with one curved surface, its skin. Its size is decided by its radius r. The radius is the distance from the centre to the surface.
surface area = 4πr²
volume = 4/3 × πr³
Cut a sphere through its centre. Each half is a hemisphere. The cut leaves a new flat face, a circle of area πr².
curved surface = 2πr²
total surface = 2πr² + πr² = 3πr²
volume = 2/3 × πr³
| Measure | Sphere | Hemisphere | Halved? |
|---|---|---|---|
| Curved surface | 5544 | 2772 | yes |
| Flat face | none | 1386 | new |
| Total surface | 5544 | 4158 | no |
| Volume | 38808 | 19404 | yes |
The curved skin and the volume halve. The total surface does not. The hemisphere gains the flat face, 1386. So 2772 + 1386 = 4158.
A cylinder with the same radius and height 2r holds the sphere. Its volume is πr² × 2r = 2πr³. The sphere's volume is two thirds of that. Check: 2/3 × 2πr³ = 4/3 πr³.