A cylinder, a hemisphere and a cone have the same base and the same height.…

For a right circular cylinder of radius r and height h, a hemisphere of radius r, and a right circular cone of radius r and slant height l, the curved surface area formulas are: cylinder = 2πrh, hemisphere = 2πr2, and cone = πrl, where the slant height satisfies l2 = r2 + h2 from the right triangle formed by the radius, height and slant height.

1. A hemisphere's own height is always equal to its radius, so the height shared by all three solids here is h = r.

2. Cylinder's curved surface area = 2πrh = 2πr·r = 2πr2.

3. Hemisphere's curved surface area = 2πr2, as in the general formula above.

4. Cone's slant height l = √(r2 + h2) = √(r2 + r2) = √(2r2) = r√2.

5. Cone's curved surface area = πrl = πr·(r√2) = √2·πr2.

6. Comparing the three areas — 2πr2 (cylinder) : 2πr2 (hemisphere) : √2·πr2 (cone) — and dividing every term by √2·πr2 gives the ratio √2 : √2 : 1.

As an independent check, take r = 1: the cylinder's area is 2π, the hemisphere's area is 2π, and the cone's area is √2·π, giving the same simplified ratio √2 : √2 : 1. This also confirms the general fact that a cylinder and a hemisphere sharing an equal base radius always have equal curved surface area (2πr2 each) whenever the cylinder's height equals that radius.

Hence, the ratio of the curved surface areas of the cylinder, the hemisphere and the cone is √2 : √2 : 1.