Formula sheet
Volume of cone
(1/3)πr²h is given on the exam formula sheet. You must still use the vertical height (not slant height), remember the 1/3 that makes a cone one-third of the matching cylinder, and use the radius.
What the symbols mean
- r Radius of the circular base (length unit).
- h Vertical height from base to apex (length unit), not the slant height.
- π Pi ≈ 3.142 or 22/7.
Given in the exam, or memorise?
(1/3)πr²h is given on the exam formula sheet. You must still use the vertical height (not slant height), remember the 1/3 that makes a cone one-third of the matching cylinder, and use the radius.
Why it works
A cone is exactly one-third of a cylinder with the same base and height.
- Take a cylinder of base πr² and height h, volume πr²h.
- A cone with the same base and height fills exactly one-third of it.
- So volume of cone = (1/3)πr²h.
Worked example 1
Find the volume of a cone with base radius 7 cm and vertical height 9 cm. (Use π = 22/7)
- Volume = (1/3)πr²h
- πr² = 22/7 × 49 = 154
- = (1/3) × 154 × 9
- = 154 × 3 = 462
Worked example 2
A cone with base radius 7 cm has volume 616 cm³. Find its vertical height.
(Use π = 22/7)
- Volume = (1/3)πr²h
- πr² = 22/7 × 49 = 154
- 616 = (1/3) × 154 × h = (154/3) h
- h = 616 × 3 ÷ 154 = 1848 ÷ 154 = 12
Where students go wrong
- Forgetting the 1/3 and getting the cylinder volume instead.
- Using the slant height in place of the vertical height h.
- Using the diameter instead of the radius, or not squaring r.
- Answering in cm² instead of cm³.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
Why is there a 1/3 in the cone volume?
Because a cone holds exactly one-third of the cylinder that has the same base and height. If you filled a cone with water three times, you would exactly fill the matching cylinder.
That is why the volume is (1/3)πr²h rather than πr²h. Dropping the 1/3 triples your answer.
Which height do I use, slant or vertical?
Volume always uses the vertical height, the straight distance from the base up to the tip. Slant height is for surface area, not volume.
If a question only gives the slant height and radius, use Pythagoras to find the vertical height first: height² = s² − r², then substitute into (1/3)πr²h.
How do I rearrange to find the height?
Set volume = (1/3)πr²h, compute πr² first, then solve for h by multiplying the volume by 3 and dividing by πr². For example volume 616 cm³ with πr² = 154 gives h = 616 × 3 ÷ 154 = 12 cm.
The ×3 undoes the 1/3, so never forget it when rearranging.