Volume of a Cone
Geometry & MeasurementThe volume of a cone is one-third times pi times the radius squared times the height.
Formula
V = \frac{1}{3}\pi r^2 h
Definition
The volume of a cone is found by multiplying $(1/3) \times \pi \times r^2 \times h$. The cone holds exactly one-third as much as a cylinder with the same base and height.
Example
An ice cream cone with radius $4$ cm and height $9$ cm: $V = (1/3) \times \pi \times 16 \times 9 = 48\pi = 150.8$ cm$^3$.
Key Insight
Fill a cone with water, then pour it into a cylinder with the same radius and height. You will need to fill and pour exactly $3$ cones to fill the cylinder. That is why there is a $1/3$ in the formula.
Definition
A cone with base radius $r$ and perpendicular height $h$ has volume $V = (1/3)\pi r^2 h$, exactly one-third of the cylinder $V = \pi r^2 h$ with the same base and height. The $1/3$ factor arises because the cross-sectional area decreases linearly from base to apex.
Example
A volcano modeled as a cone: base radius $5$ km, height $3$ km. $V = (1/3)\pi \cdot 25 \cdot 3 = 25\pi = 78.5$ km$^3$. Doubling only the height doubles the volume; doubling only the radius quadruples it.
Key Insight
The factor of $1/3$ in the cone formula is not an accident: all shapes that taper to a point (pyramids and cones) have $1/3$ times the volume of the corresponding prism. This is a universal consequence of how cross-sectional area changes linearly from base to apex.
Definition
$$V = \int_0^h \pi \left(\frac{rz}{h}\right)^2 dz = \frac{\pi r^2}{h^2} \cdot \frac{h^3}{3} = \frac{1}{3}\pi r^2 h.$$ The cross-section at height $z$ is a disk of radius $r(z/h)$ (linear scaling from $0$ at apex to $r$ at base). The integral of a quadratic function over $[0,h]$ produces the factor $1/3$.
Example
For an oblique cone (apex not above center), the same formula applies with $h$ as the perpendicular height from apex to base plane. This follows from Cavalieri's principle: each horizontal cross-section has the same area as in the right cone at the same height.
Key Insight
The $1/3$ factor is equivalent to integrating $x^2$ from $0$ to $1$, which gives $1/3$. This is why all "tapering to a point" solids have the $1/3$ factor: the cross-sectional area scales as the square of the distance from the apex, and integrating $x^2$ always yields $1/3$.