Final answer:
The surface area of a cone is derived by summing the area of the base (πr^2) and the lateral surface area (πrl), where r is the radius of the base and l is the slant height.
Step-by-step explanation:
The derivation of the surface area of a cone consists of finding the sum of the area of the circular base and the lateral (curved) surface area.
To start with, the area of the circular base is simple and given by the formula πr2.
The lateral surface area is a bit more complex to derive, as it can be represented as a sector of a larger circle (the cone's development) with a radius equal to the slant height of the cone, l.
The length of the arc of this sector is the circumference of the base of the cone, 2πr,
so the area of the sector (and the lateral surface area of the cone) is given by the proportion (lateral surface area/total area of the larger circle) = (circumference of the cone base)/(circumference of the larger circle), which simplifies to πrl.
Thus, the total surface area of the cone is πr2 + πrl.