Volume of a Cone — Formula, Derivation & Examples

The volume of a cone is the amount of space inside it. For a cone with base radius $r$ and perpendicular height $h$, use

$$V = \frac{1}{3}\pi r^2 h$$

The base area is $\pi r^2$, so this formula is really saying:

$$\text{cone volume} = \frac{1}{3}(\text{base area})(\text{height})$$

If the base is circular, that becomes $\frac{1}{3}\pi r^2 h$.

What the formula means

The factor $\pi r^2$ is the area of the circular base. Multiplying by $h$ would give the volume of a cylinder with the same base and height.

A cone is more tapered, so it holds less than that cylinder. In fact, with the same base area and height, it holds exactly one-third as much.

That gives the fastest intuition for the formula:

$$V_{\text{cone}} = \frac{1}{3}V_{\text{cylinder}} = \frac{1}{3}\pi r^2 h$$

Why the one-third appears

One standard derivation uses cross-sections. Measure height $x$ upward from the tip of the cone. At that level, the radius scales linearly, so

$$\text{radius at height } x = \frac{r}{h}x$$

The cross-sectional area there is

$$A(x) = \pi \left(\frac{r}{h}x\right)^2$$

Add those thin circular slices from $x = 0$ to $x = h$:

$$V = \int_0^h \pi \left(\frac{r}{h}x\right)^2 dx$$ $$V = \frac{\pi r^2}{h^2}\int_0^h x^2 dx = \frac{\pi r^2}{h^2}\left(\frac{h^3}{3}\right) = \frac{1}{3}\pi r^2 h$$

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

One worked example

Suppose a cone has radius $3$ cm and height $8$ cm.

Start with the formula:

$$V = \frac{1}{3}\pi r^2 h$$

Substitute $r = 3$ and $h = 8$:

$$V = \frac{1}{3}\pi (3^2)(8)$$

Square the radius and simplify:

$$V = \frac{1}{3}\pi (9)(8) = 24\pi$$

So the exact volume is

$$24\pi\ \text{cm}^3$$

If you need a decimal approximation,

$$24\pi \approx 75.4\ \text{cm}^3$$

Common mistakes

1. Using the diameter as if it were the radius. If the diameter is $6$ cm, the radius is $3$ cm.
2. Using slant height instead of perpendicular height. The volume formula needs the straight height from base to tip.
3. Forgetting the one-third factor. $\pi r^2 h$ is the cylinder formula, not the cone formula.
4. Forgetting to square the radius. The formula uses $r^2$, not $r$.
5. Dropping cubic units. Volume should be written in units such as $\text{cm}^3$ or $\text{m}^3$.

When the formula is used

This formula is used in geometry, engineering estimates, packaging, and any problem where a shape can be modeled as a cone or near-cone. Common examples include funnels, piles of material, and conical tanks.

If the object is only approximately conical, the result is also an approximation. The closer the shape is to a true cone, the more useful the estimate will be.

A quick check that catches errors

If a cone and a cylinder have the same base radius and the same height, the cone's volume should be smaller by a factor of $3$.

So if your cone answer comes out equal to $\pi r^2 h$, you have probably missed the $\frac{1}{3}$.

Try your own version

Try your own version with 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. If you want to compare a few cases quickly, solve a similar problem with GPAI Solver.