Cone
Geometry & MeasurementA cone is a 3-D solid with a circular base tapering to a single point called the apex.
Formula
V = \frac{1}{3}\pi r^2 h; \ LSA = \pi r l
Definition
A cone has a circular base and comes to a single point at the top called the apex or vertex. An ice cream cone and a traffic cone are real-life examples.
Example
An ice cream cone with radius $3$ cm and height $12$ cm: Volume $= (1/3) \times \pi \times 9 \times 12 = 36\pi = 113.1$ cm$^3$. The cone holds exactly one-third as much as a cylinder with the same base and height.
Key Insight
Three cones of the same size can be filled with sand and poured into one cylinder of the same base and height. They fill it exactly! That is why the cone volume formula has the "one-third."
Definition
A right circular cone has a circular base of radius $r$, a perpendicular height $h$ from center of base to apex, and a slant height $l = \sqrt{r^2 + h^2}$. Volume $V = (1/3)\pi r^2 h$; Lateral surface area $LSA = \pi r l$; Total $SA = \pi r^2 + \pi r l = \pi r(r + l)$.
Example
Cone: $r = 5$ cm, $h = 12$ cm. Slant height $l = \sqrt{25 + 144} = \sqrt{169} = 13$ cm. $V = (1/3)\pi \cdot 25 \cdot 12 = 100\pi = 314.2$ cm$^3$. $LSA = \pi \cdot 5 \cdot 13 = 65\pi = 204.2$ cm$^2$.
Key Insight
The slant height $l$ is the distance along the surface from the apex to the base edge, not the vertical height $h$. Confusing $l$ and $h$ is a common error. $l$ is always longer than $h$ (it is the hypotenuse of the right triangle formed by $r$ and $h$).
Definition
A cone is the solid of revolution of a right triangle about one leg, or equivalently the convex hull of a disk and an apex point. Volume $V = (1/3)\pi r^2 h$ is derived by integrating circular cross-sections: $$V = \int_0^h \pi \left(\frac{rz}{h}\right)^2 dz = \frac{\pi r^2}{h^2} \cdot \frac{h^3}{3} = \frac{1}{3}\pi r^2 h.$$
Example
Cavalieri's principle: a cone and a hemisphere of equal radius $r$ fit together inside a cylinder of radius $r$ and height $r$. At height $z$, the cone cross-section has area $\pi (rz/h)^2$ and the hemisphere cross-section has area $\pi(r^2-z^2)$. Their sum equals $\pi r^2$ (the cylinder), proving $V_{cone} + V_{hemisphere} = V_{cylinder}$.
Key Insight
This elegant Archimedean proof shows $V_{hemisphere} = (2/3)\pi r^3$ and $V_{cone} = (1/3)\pi r^3$ without calculus. The method of indivisibles (Cavalieri's principle) is a precursor to integral calculus and one of the most beautiful proofs in classical geometry.