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Theorem — what does that mean?
Half of the sphere: two thirds of pi times the radius cubed.
In Swedish: Halvklotets volym
V
Half of the sphere: two thirds of pi times the radius cubed.
V≈56.5487cm³
V equals ((2 times π) times 3 cubed) divided by 3, is approximately (6.2832 times 27) divided by 3, is approximately 169.646 divided by 3, is approximately 56.5487 cm³
You are given
Round to 1 decimal.
Pick the unit your lengths are in. Areas are then labelled in square units and volumes in cubic units; nothing is converted.
r is the radius. A sphere holds 4πr³ / 3; a hemisphere is half of it, so the 4 becomes a 2. A bowl, a dome and a cut ball are all hemispheres if the cut runs through the centre.
A radius of 3 cm: V = 2π · 27 / 3 = 18π ≈ 56.5 cm³. A dome with a radius of 6 m encloses 2π · 216 / 3 = 144π ≈ 452 m³: double the radius, eight times the volume.
Turned round, r = ∛(3V / 2π). A bowl that holds 1 litre, 1,000 cm³, has a radius of ∛(3,000 / 2π) = ∛477 ≈ 7.8 cm. The card above does the cube root.
Volume grows with the cube of the radius, so a bowl 10% wider holds 1.1³ = 1.33 times as much, a third more. That is why a small change in the radius is a large change in what the bowl holds, and why a volume rounded to the nearest centimetre of radius can be off by a tenth.
What is the surface area?
3πr², the curved half plus the flat circle, on the surface area of a hemisphere page.
How does it compare with a cone of the same radius and height?
A cone with height r holds πr³ / 3, half the hemisphere; a cylinder of the
same size holds πr³, one and a half times it. The three are in the ratio 1 : 2 : 3.
Does the formula work for a cap that is not a half?
No. A cap cut above or below the centre has its own formula with the cap’s height in it.
| Solve for | Rearranged | With the numbers | Comes to |
|---|---|---|---|
| r | 3 | 3 | 3 cm |
| V | 2πr³3 | 2π·3³3 | 56.5 cm³ |
Last reviewed October 20, 2026 ·How pages are reviewed
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