The volume of a sphere is V = 4/3 πr³, where r is the radius. The formula is on the GCSE formula sheet, but you still need to identify r from the question — it may be given as a diameter — and decide whether to leave the answer in terms of π or evaluate it as a decimal.
What is the formula for the volume of a sphere?
V = (4/3)πr³
Every part of the formula has a purpose:
- 4/3 — a constant that comes from the integration that derives the formula; it is irrational and cannot be simplified further.
- π — the ratio of a circle's circumference to its diameter, approximately 3.14159.
- r³ — the cube of the radius, reflecting that volume is a three-dimensional quantity.
The formula is listed on the GCSE Higher and Foundation formula sheets. You are expected to recall how to identify r correctly and apply the formula, not to derive it from first principles.
How do you identify the radius from the question?
The most common mistake on sphere questions is confusing radius and diameter. Always check:
- If the question gives the diameter (d), the radius is r = d/2.
- If the question gives the radius directly, use it straight away.
Example: A sphere has diameter 10 cm. What is r? r = 10/2 = 5 cm.
If you cube the diameter instead of the radius you will get 8 times the correct answer, losing all the marks.
How do you calculate the volume of a sphere step by step?
Worked example 1: Find the volume of a sphere of radius 6 cm. Give your answer to 3 significant figures.
- Identify r = 6 cm.
- Apply the formula: V = (4/3) × π × 6³.
- Calculate 6³ = 216.
- V = (4/3) × π × 216 = (4 × 216)/3 × π = 288π.
- Evaluate: V = 288 × 3.14159… = 904 cm³ (to 3 s.f.).
Worked example 2: Find the volume of a sphere of diameter 9 m. Give your answer in terms of π.
- r = 9/2 = 4.5 m.
- V = (4/3) × π × (4.5)³.
- (4.5)³ = 4.5 × 4.5 × 4.5 = 20.25 × 4.5 = 91.125.
- V = (4/3) × 91.125 × π = 4 × 91.125/3 × π = 364.5/3 × π = 121.5π m³.
What is the volume of a hemisphere?
A hemisphere is exactly half a sphere. Its volume is simply half the full sphere's volume:
V (hemisphere) = (1/2) × (4/3)πr³ = (2/3)πr³
Worked example: Find the volume of a hemisphere of radius 3 cm, in terms of π.
V = (2/3) × π × 3³ = (2/3) × 27π = 18π cm³.
Some questions combine a hemisphere with a cylinder or cone (a "composite solid"). Find the volumes of each part separately, then add them together.
How do you find the radius when the volume is given?
Rearrange the formula to isolate r.
Starting from V = (4/3)πr³:
- Multiply both sides by 3/4: (3V)/(4π) = r³.
- Take the cube root: r = ∛(3V/4π).
Worked example: A sphere has volume 113 cm³. Find its radius to 1 decimal place.
- r³ = (3 × 113)/(4π) = 339/(4π) = 339/12.566 ≈ 26.98.
- r = ∛26.98 ≈ 3.0 cm (to 1 d.p.).
Check: V = (4/3)π(3)³ = (4/3) × π × 27 = 36π ≈ 113. ✓
Common errors to avoid
| Error | Example of error | Correct approach |
|---|---|---|
| Using diameter instead of radius | V = (4/3)π × 10³ when d = 10 | First halve: r = 5, then cube |
| Not cubing the radius | V = (4/3)π × 6 | r³ = 6³ = 216, not 6 |
| Rounding π too early | Using 3.14 then rounding again | Keep full calculator value of π until the final step |
| Wrong hemisphere formula | Halving the formula incorrectly | V = (2/3)πr³, not (2/3)πr² |
Frequently asked questions
Is the formula for the volume of a sphere on the GCSE formula sheet?
Yes — V = (4/3)πr³ is provided on both the AQA and Edexcel GCSE Higher formula sheets. You do not need to memorise it, but you do need to practise applying it correctly, including recognising r from a diameter and evaluating r³ without error.
How is the volume of a sphere related to the surface area?
The surface area of a sphere is A = 4πr². This is the formula for the surface area, not the volume — confusing the two is a common error. Note that 4πr² is the derivative of (4/3)πr³ with respect to r, a beautiful calculus relationship, but at GCSE you simply use each formula in the appropriate context.
What units does the volume come out in?
Volume is always in cubic units. If the radius is in cm, the volume is in cm³. If the radius is in m, the volume is in m³. If a question mixes units (e.g. a radius in mm but you want the volume in cm³), convert the radius to the target unit first, then apply the formula.
Can you use the formula for a sphere to find the volume of a cone or cylinder?
No — each solid has its own formula. The volume of a cone is (1/3)πr²h; the volume of a cylinder is πr²h. A useful memory check: the volume of a sphere with radius r equals exactly (4/3) times the volume of a cone with the same radius and height 2r — but at GCSE you apply each formula separately from the formula sheet.
For 3D shape and volume practice at GCSE with Professor Pi, visit aitutors.me.