MQ11 --- Mensuration

The frustum is made by cut off a small cone withbase radius x from a large cone with base radius y,
and the difference between the slant heights of the two cones is x.

Let h be the slant height of the small cone.
Then, the slant height of the large cone = h+z

The two cones are similar solids.
(h+z)/h = y/x
hy = hx+xz
h(y-x) = xz
h = xz/(y-x)

The height of the large cone
= [xz/(y-x)] + z
= (xz+yz-zx)/(y-x)
= yz/(y-x)

Consider the small cone :
Base radius = x
Slant height = xz/(y-x)
Curved surface area = (1/2)*[xz/(y-x)]*(2*π*x) = πx²z/(y-x)

Consider the large cone :
Base radius = y
Slant height = yz/(y - x)
Curved surface area = (1/2)*[yz/(y - x)]*(2*π*y) = πy²z/(y - x)

Curved surface area of the frustum
= (Curved surface area of the large cone) - (Curved surface area of the smallcone)
= [πy²z/(y-x)] - [πx²z/(y-x)]
= πz(y²-x²)/(y-x)
= πz(y-x)(y+x)/(y-x)
= πz(x+y)