Cone
A 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, like an ice cream cone or traffic cone; for a right circular cone with base radius $r$, perpendicular height $h$, and slant height $l = \sqrt{r^2 + h^2}$, volume $V = (1/3)\pi r^2 h$, lateral surface area $LSA = \pi r l$, and total $SA = \pi r(r + l)$. It is the solid of revolution of a right triangle about one leg, and its volume can be 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
An ice cream cone with radius $3$ cm and height $12$ cm has volume $(1/3) \times \pi \times 9 \times 12 = 36\pi = 113.1$ cm$^3$, holding exactly one-third as much as a cylinder with the same base and height. A cone with $r = 5$ cm, $h = 12$ cm has slant height $l = \sqrt{25 + 144} = \sqrt{169} = 13$ cm, so $V = (1/3)\pi \cdot 25 \cdot 12 = 100\pi = 314.2$ cm$^3$ and $LSA = \pi \cdot 5 \cdot 13 = 65\pi = 204.2$ cm$^2$. Cavalieri's principle shows a cone and a hemisphere of equal radius $r$ fit together inside a cylinder of radius $r$ and height $r$, since at height $z$ the cone cross-section area $\pi (rz/h)^2$ plus the hemisphere cross-section area $\pi(r^2-z^2)$ sum to $\pi r^2$, proving $V_{cone} + V_{hemisphere} = V_{cylinder}$.
Key Insight
Three cones of the same size can be filled with sand and poured into one cylinder of the same base and height, filling it exactly, which is why the volume formula has the "one-third." The slant height $l$ is the distance along the surface from apex to base edge, not the vertical height $h$, and confusing the two is a common error; $l$ is always longer than $h$, the hypotenuse of the right triangle formed by $r$ and $h$. Archimedes' elegant proof shows $V_{hemisphere} = (2/3)\pi r^3$ and $V_{cone} = (1/3)\pi r^3$ without calculus, a precursor to integral calculus.