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No, it never can equal one third of the lateral surface area:
If the base of the cylinder and cone has radius r, and the height of the cone and cylinder has height h, then:
Lateral surface area of a cone = πr√(r2+h2)
Lateral surface area of a cylinder = 2πrh
The lateral surface area of a cone equals one third the lateral surface area of a cone when:
πr√(r2+h2) = 1/3 x 2πrh
⇒ √(r2+h2) = 2/3h
⇒ r2+h2 = 4/9h2
⇒ r2 = -5/9h2
But a square number can never be negative, so this is impossible.
However, the volume of a cone is one third the volume of the cylinder with the same radius r and height h:
Volume cone = 1/3πr2h
Volume cylinder = πr2h
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