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 r/s × pi×s^2 = pi×r×s. Add the base circle, pi×r^2.

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

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