Cone: Slant, Surface Area, Volume

A cone of base radius r and height h has slant height $l = \sqrt{r^2 + h^2}$, curved surface $CSA = \pi r l$, total surface $TSA = \pi r(r + l)$, and volume $V = \frac{1}{3}\pi r^2 h$. The volume is exactly one-third of the cylinder with the same base and height.

Slide r and h; l, CSA, TSA and V all update together. Try r = 6, h = 8 to see slant l = 10, CSA = 60π and volume = 96π. Notice how slant uses Pythagoras.

Students forget the $\frac{1}{3}$ in the cone volume formula and end up with the cylinder volume. They also confuse height h with slant height l in the CSA formula. Always compute l first using $\sqrt{r^2 + h^2}$.

An ice-cream cone, a traffic cone, and a heap of rice on a thali are all cone-shaped. A wheat-grain heap of radius 2 m and height 1.5 m holds $\frac{1}{3}\pi \cdot 4 \cdot 1.5$ cubic metres - useful at harvest time.