Sheet № 105 · Higher only · AQA · Edexcel · OCR
Volume of a Sphere –
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.
§Key definitions
Question:
A sphere has a radius of 6 cm. Find its volume in terms of pi.
Answer:
288pi cm³
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.
§Formulas to memorise
V = ⁴⁄₃ pi r³, where r is the radius of the sphere
Volume of a hemisphere = ²⁄₃ pi r³
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
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
⚠ 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.