If the curved surface area of a solid cylinder is one-third of its total surface area, then what is the ratio of the diameter to the height?

4 : 1

C.S.A. = 2\(\pi \)rh

T.S.A. = C.S.A. + 2\(\pi \)r²

ATQ,

T.S.A. : C.S.A. = 3 : 1

C.S.A. + 2\(\pi \)r² : C.S.A. = 3 :1

⇒ 2\(\pi \)r²  : 2\(\pi \)rh = 2 : 1                   (C.S.A. = 1R)

⇒ r : h = 2 : 1

or 2r : h = 4 : 1

Hence,

Diameter : Height = 4 : 1