Learn: a ball and a half-ball mould
Picture a solid ball and a half-ball mould with the same radius. A hemisphere is half a sphere cut through its centre. Its volume is exactly half the full sphere’s volume.
The radius r is the distance from the centre to the surface. The diameter d = 2r passes through the centre from one side to the other. Volume V measures enclosed space, in cubic units. π (pi) is approximately 3.14159; r³ means r × r × r.
Sphere: V = (4/3)πr³.
Hemisphere: V = (2/3)πr³.
The hemisphere’s flat circular face has area, but no thickness and therefore no volume to add. If you are finding the amount of material in a thick bowl, its walls require a separate calculation. Do not add πr² to a volume: square and cubic units represent different quantities.
A useful comparison
A sphere of radius r fits inside a cylinder of radius r and height 2r. That cylinder’s volume is 2πr³. The sphere’s volume is two thirds of this cylinder’s volume. This comparison helps check magnitude; it is not a full derivation of the sphere formula.
Worked example: a sphere of radius 6 cm
- Cube the radius: 6³ = 216 cm³.
- Multiply: V = (4/3)π × 216 = 288π cm³.
- Approximate at the end: V ≈ 904.78 cm³.
A hemisphere of the same radius has volume 144π cm³ ≈ 452.39 cm³. The two halves together recover the sphere.
Worked example: a hemispherical mould of diameter 8 cm
Its inside radius is 4 cm. The capacity is V = (2/3)π × 4³ = (128/3)π cm³ ≈ 134.04 cm³, or about 134.04 mL. This assumes a true hemispherical interior filled to its flat rim.
Why doubling the radius matters
Replacing r with 2r multiplies r³ by 8. Both sphere and hemisphere volumes become eight times as large. A shape that is twice as wide in every direction holds eight times the space, not twice.
Explore: change one dimension
The diagram is schematic, not to scale. Quantities below it update when you move the sliders. Predict the change first, then compare the result.
Try radius 3 cm, then 6 cm. For volume, keep the height fixed where available. Explain why the change differs from simply doubling the result. The shape drawing stays fixed so you can focus on the labelled dimensions and numerical comparison.
Practice: explain before checking
Work each question on paper. Open the explanation to compare your setup, calculation and units.
1. Find the volume of a hemisphere with radius 3 m.
V = (2/3)π × 27 = 18π m³ ≈ 56.55 m³. The matching sphere is 36π m³.
2. A sphere has diameter 10 ft. Which radius belongs in the formula?
Use r = 5 ft. V = (4/3)π × 125 = (500/3)π ft³ ≈ 523.60 ft³. Using 10 ft as the radius would give eight times too much.
3. Do you add the circular base when finding hemisphere volume?
No. A face has area but no volume. The hemisphere’s volume is exactly one half of the full sphere’s volume.
Review
Halve a given diameter, cube the radius, use the correct fraction and report cubic units. A hemisphere’s flat face adds no volume.
Recall without looking: What does each symbol measure? Which units belong in the answer? Which surfaces or spaces are included?