Volume of a Cone — Formula, Derivation & Examples

Reach for the cone volume formula whenever an object narrows from a circular base to a single point: funnels, piles of sand, paper cups, conical tanks. As long as the shape is a true cone or close to one, the same procedure gives the space inside it. For base radius $r$ and perpendicular height $h$,

Since the base area is $\pi r^2$, the formula is really saying

The Procedure, Step By Step

1. Identify the base radius $r$. If you are given the diameter, divide by $2$ first.
2. Identify the perpendicular height $h$, measured straight from base to tip, not the slant height.
3. Substitute into $V = \frac{1}{3}\pi r^2 h$, squaring the radius before multiplying.
4. Keep the units consistent so the final volume comes out in cubic units.
5. Decide on exact or decimal form, leaving the answer in terms of $\pi$ unless a decimal is requested.

Why The One-Third Is There

Multiplying base area $\pi r^2$ by height $h$ alone would give the volume of a cylinder with the same base and height. A cone is more tapered, so it holds less, exactly one-third as much:

One standard derivation makes that precise with cross-sections. Measure height $x$ upward from the tip; the radius scales linearly, so

The cross-sectional area there is

Adding those thin slices from $x = 0$ to $x = h$,

If you have not studied calculus yet, the practical takeaway still holds: a cone with the same base and height as a cylinder has one-third the volume.

A Worked Run Through The Steps

Take a cone with radius $3$ cm and height $8$ cm.

The radius is $r = 3$ and the perpendicular height is $h = 8$. Substitute:

Square the radius and simplify:

So the exact volume is

and as a decimal,

Where Each Step Goes Wrong, And How To Catch It

• Reading the radius: using the diameter as if it were the radius. A diameter of $6$ cm means a radius of $3$ cm.
• Reading the height: using slant height instead of the perpendicular height from base to tip.
• Substituting: forgetting to square the radius. The formula uses $r^2$, not $r$.
• Substituting: dropping the one-third. Note that $\pi r^2 h$ is the cylinder formula, not the cone formula.
• Labeling: leaving off cubic units such as $\text{cm}^3$ or $\text{m}^3$.

A self-check that catches most of these: a cone and cylinder with the same radius and height differ by a factor of $3$. If your cone answer comes out equal to $\pi r^2 h$, you have probably missed the $\frac{1}{3}$. And remember the model condition: if the object is only approximately conical, the result is also an approximation.

Run It Yourself

Work the same five steps for radius $5$ and height $12$. Before calculating, predict whether the exact answer should be larger or smaller than the cylinder volume with the same dimensions. To compare several cases quickly, solve a similar problem with GPAI Solver.