Cone volume and surface area
Measure a right circular cone’s volume and total surface area.
Cone volume and surface area is one of 8 solids formulas in the geometry section of this library, and it is used at high school level.
Why cone volume and surface area works
A cone is the round relative of a pyramid, so it holds a third of the cylinder that just contains it. The curved wall, when slit and flattened, becomes a sector of a circle whose radius is the slant length, and that sector works out to π times the base radius times the slant.
What each symbol means
$r$ is radius, $h$ vertical height, and $\ell$ slant height.
Cone volume and surface area: when it holds
$\ell=\sqrt{r^2+h^2}$ for a right cone; include the base for total area.
When it stops applying
The relation between radius, height and slant holds only for an upright cone whose apex is directly above the centre of the base. In a leaning cone the slant differs all the way around the rim, so the curved-surface expression is meaningless even though the volume survives.
Cone volume and surface area: a worked example
For $r=3,h=4,\ell=5$, $V=12\pi$ and $S=24\pi$.
The mistake to avoid
What people do: Putting the slant length into the volume expression.
Why it goes wrong: Volume needs the vertical height; with a base radius of 3, using a slant of 5 in place of the height 4 reports 15π instead of the correct 12π.
Do this instead: Keep the two apart: the vertical height belongs to the volume and the slant belongs to the curved surface.
Cone volume and surface area: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $r$ is radius, $h$ vertical height, and $\ell$ slant height.
- Check the conditions before substituting. $\ell=\sqrt{r^2+h^2}$ for a right cone; include the base for total area.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Solids
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where cone volume and surface area comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about cone volume and surface area
How do I find the slant length?
Use the right triangle formed by the base radius and the vertical height. A radius of 3 with a height of 4 gives a slant of 5.
What is the curved surface on its own?
It is π times the base radius times the slant length. Leave the base disk out whenever the cone is open, like a paper party hat.
How does a cone compare with a cylinder of the same size?
It holds exactly one third as much, so three cone-loads of sand are needed to fill a cylinder of the same radius and height.
Why does the curved wall flatten into a sector?
Every point on the rim sits the same slant distance from the apex, so flattening the wall produces part of a circle centred on that apex.