Formula sheet
Volume of sphere
(4/3)πr³ is given on the exam formula sheet. You must still cube the radius (not square it), keep the 4/3, use the radius not the diameter, and give the answer in cubic units.
What the symbols mean
- r Radius of the sphere (length unit). Halve the diameter if only d is given.
- π Pi ≈ 3.142 or 22/7.
Given in the exam, or memorise?
(4/3)πr³ is given on the exam formula sheet. You must still cube the radius (not square it), keep the 4/3, use the radius not the diameter, and give the answer in cubic units.
Why it works
The sphere volume relates to its surface area and radius.
- Think of the surface (area 4πr²) as made of tiny cones reaching the centre.
- Each has height r, and their volumes total (1/3) × surface × r.
- That is (1/3) × 4πr² × r = (4/3)πr³.
Worked example 1
Find the volume of a sphere of radius 21 cm. (Use π = 22/7)
- Volume = (4/3)πr³
- r³ = 21³ = 9261
- 22/7 × 9261 = 22 × 1323 = 29106
- = (4/3) × 29106 = 4 × 9702 = 38808
Worked example 2
Find the volume of a sphere of radius 10.5 cm. (Use π = 22/7)
- Volume = (4/3)πr³
- r³ = 10.5³ = 1157.625
- 22/7 × 1157.625 = 22 × 165.375 = 3638.25
- = (4/3) × 3638.25 = 4 × 1212.75 = 4851
Where students go wrong
- Squaring r instead of cubing it, volume needs r³, surface area needs r².
- Forgetting the 4/3 factor.
- Using the diameter in place of the radius.
- Answering in cm² instead of cm³.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
How do I tell the sphere volume and surface area apart?
Volume is (4/3)πr³ in cm³, with the radius cubed. Surface area is 4πr² in cm², with the radius squared.
The power of r and the unit go together: cube with cm³ is volume, square with cm² is surface area. Reading the unit the question asks for tells you which formula to reach for.
What if I only know the diameter?
Halve it to get the radius before substituting, because (4/3)πr³ uses r. A ball of diameter 21 cm has r = 10.5 cm, so cube 10.5, not 21.
Since the radius is cubed, using the diameter would make the volume eight times too large. Always halve first, then cube.
How do I cube a decimal radius like 10.5 cleanly?
Cube it in steps: 10.5² = 110.25, then 110.25 × 10.5 = 1157.625. Keep all decimals until the end to avoid rounding errors, then round the final volume if the question asks.
Choosing π = 22/7 with a radius that is a multiple of 3.5 or 7 often makes the arithmetic tidy.