Surface of a Cone

Cut a cone's curved side along its slant and flatten it: you get a slice of a circle whose radius is the slant height s. Its curved edge is the rim of the base, 2πr long, out of the 2πs of a whole circle of radius s. So the slice is the fraction r/s of that circle, and its area is $\frac{r}{s}\cdot \pi\cdot {s}^{2} = \pi\cdot r\cdot s$. Add the base circle, $\pi\cdot {r}^{2}$.

This cone has radius 3 and slant height 5. The base is $\pi\cdot {3}^{2} = 9\cdot \pi$, the curved side is $\pi\cdot 3\cdot 5 = 15\cdot \pi$, and the total is 24π.

The rule is Surface area of a cone: $SA = \pi\cdot {r}^{2}+\pi\cdot r\cdot s$. Like the pyramid, the curved side uses the slant height, never the true height.