Volume of Spheres and Hemispheres: Formula & Examples

A sphere is a perfectly round ball — a football, a marble or a planet — and a hemisphere is exactly half of one, like a bowl or a dome. Their volumes depend on a single number, the radius, but the sphere's formula has a 4/3 factor that surprises many students: V = (4/3)πr³ for the full sphere and V = (2/3)πr³ for the hemisphere. The power of three makes sense because volume is three-dimensional, so the radius appears three times.

Spherical volumes appear wherever balls and domes are filled or built. How much air goes into a basketball, how much water a hemispherical bowl holds, how much concrete forms a dome roof — all of these are sphere or hemisphere volume problems. The hemisphere is also half the fun of capsule shapes, from medicine capsules to toy tops.

Key Concepts

• Radius (r): the distance from the centre of the sphere to any point on its surface.
• The 4/3 factor: a sphere's volume is four-thirds of πr³; the hemisphere, being half a sphere, uses two-thirds of πr³.
• Cubing the radius: r³ = r × r × r — all three copies of the radius must be multiplied.
• Volume units: cubic units (cm³, m³); 1000 cm³ = 1 litre.
• The radius must be in the same unit throughout the calculation.

Formulas

• Volume of a sphere: V = (4/3)πr³.
• Volume of a hemisphere: V = (2/3)πr³ — exactly half of the sphere's volume.
• Finding the radius from the volume: r³ = 3V ÷ (4π), then take the cube root.
• These assume a solid sphere; a hollow spherical shell needs the outer volume minus the inner volume, (4/3)π(R³ − r³).

Worked Example

Find the volume of a sphere of radius 3 cm. Use π = 22 ÷ 7.

1. Write the formula: V = (4/3) × π × r³ = (4/3) × (22 ÷ 7) × 27.
2. Compute r³ = 3 × 3 × 3 = 27.
3. Multiply: (4/3) × (22 ÷ 7) × 27 = 4 × 22 × 27 ÷ (3 × 7) = 2376 ÷ 21 ≈ 113.14. The volume is about 113.14 cm³.
4. Bonus — a hemispherical bowl of radius 7 cm: V = (2/3) × (22 ÷ 7) × 343 = (2/3) × 22 × 49 = 2156 ÷ 3 ≈ 718.67 cm³, or about 0.72 litres.

Common Misconceptions

• Using r² instead of r³: volume needs the radius cubed; r² gives an area, not a volume.
• Forgetting the 4/3: πr³ alone is not a sphere's volume; the full factor (4/3) must be included.
• Using the diameter as r: if a ball's diameter is 6 cm, the radius is 3 cm — halving before cubing avoids an eightfold error, since the error gets cubed too.

Real-World Applications

• Inflating sports balls, balloons and airships to the right pressure and size.
• Designing hemispherical bowls, cooking vessels and storage domes.
• Calculating concrete for dome roofs and planetarium shells.
• Estimating the capacity of spherical water tanks and gas holders.

A sphere holds (4/3)πr³ and a hemisphere holds (2/3)πr³ — cube the radius, never square it, and never drop the fraction. Halve any diameter first, keep the units cubic, and divide by 1000 to convert cm³ into litres.