The volume of a sphere is a Higher-tier formula that appears frequently in GCSE exams across AQA, Edexcel and OCR. It is provided on the formula sheet, so the challenge is applying it correctly — especially when dealing with hemispheres or working backwards to find the radius from a given volume.
What Is the Volume of a Sphere?
A sphere is a perfectly round 3D shape where every point on the surface is the same distance from the centre. That distance is the radius.
The volume formula involves cubing the radius and multiplying by four-thirds of pi. This means even small changes in the radius have a large effect on the volume — doubling the radius multiplies the volume by eight.
A hemisphere is exactly half of a sphere, so its volume is half the sphere volume. Hemisphere questions are very common at GCSE and are a simple extension of the sphere formula.
Key Formulas
Step-by-Step Method
- Check whether you have the radius or the diameter. If given the diameter, halve it.
- Substitute into V = ⁴⁄₃ pi r³ and calculate.
- For a hemisphere, use V = ²⁄₃ pi r³ (or halve the sphere volume).
Worked Example 1 — Foundation Level
Question: A sphere has a radius of 6 cm. Find its volume in terms of pi.
Working: V = ⁴⁄₃ pi r³ V = ⁴⁄₃ × pi × 6³ V = ⁴⁄₃ × pi × 216 V = 864 ÷ 3 × pi V = 288pi
Answer: 288pi cm³
Worked Example 2 — Higher Level
Question: A sphere has a volume of 500 cm³. Find its radius to 2 decimal places.
Working: V = ⁴⁄₃ pi r³ 500 = ⁴⁄₃ pi r³ r³ = 500 × 3 ÷ (4pi) r³ = 1500 ÷ (4pi) r³ = 1500 ÷ 12.566... r³ = 119.366... r = cube root of 119.366... r = 4.922...
Answer: 4.92 cm (2 d.p.)
Worked Example 3 — Exam Style
Question: A solid hemisphere has a radius of 5 cm. Find its volume to 1 decimal place.
Working: Full sphere volume = ⁴⁄₃ × pi × 5³ = ⁴⁄₃ × pi × 125 = 500pi ÷ 3 = 166.666... × pi Hemisphere volume = ½ × 500pi ÷ 3 = 250pi ÷ 3 = 261.799...
Answer: 261.8 cm³ (1 d.p.)
Common Mistakes
- Confusing r³ with r². The sphere formula uses r cubed, not r squared. Squaring instead of cubing gives a surface-area-like answer.
- Using diameter instead of radius. The formula uses r. If given a diameter of 10 cm, the radius is 5 cm — using 10 will give a volume eight times too large.
- Forgetting to halve for hemispheres. A hemisphere is half a sphere. Divide the full sphere volume by 2.
Exam Tips
- The formula is on the formula sheet. Copy it into your working and show the substitution clearly.
- To find the radius from a given volume, rearrange to r³ = 3V ÷ (4pi), then cube-root the result.
- When a sphere and hemisphere appear in the same question (e.g. a composite solid), calculate each separately and combine at the end.
- Use estimation to check: a sphere with radius 10 cm has volume roughly 4189 cm³ (since ⁴⁄₃ × pi × 1000 is about 4189). If your answer is vastly different, recheck your working.
Practice Questions
Q1 (Foundation): A sphere has a radius of 3 cm. Find its volume to 1 decimal place.
Q2 (Foundation): A hemisphere has a radius of 8 cm. Find its volume to the nearest whole number.
Q3 (Higher): A sphere has a volume of 972pi cm³. Find the exact radius.
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Related Topics
Summary
- The volume of a sphere is V = ⁴⁄₃ pi r³.
- A hemisphere has half the sphere volume: V = ²⁄₃ pi r³.
- Always use the radius, not the diameter.
- To find the radius from volume, rearrange to r³ = 3V ÷ (4pi) and cube-root.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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