Geometry

Solids

Medium

If the ratio of the radii of the sphere and hemisphere is √3:2, then determine the ratio of their total surface area?

1:2

2:1

1:3

1:1

✅ Correct Option: 4

The ratio of radii is Sphere : Hemisphere $= \sqrt{3} : 2$ .

Let the radius of sphere $R_1 = \sqrt{3}k$ and radius of hemisphere $R_2 = 2k$ , where $k$ is a constant.

The total surface area of a sphere is $4\pi r^2$ .

The total surface area of a hemisphere includes the curved surface and the flat circular base: $2\pi r^2 + \pi r^2 = 3\pi r^2$ .

Total surface area of sphere:

$\text{TSA}_{\text{sphere}} = 4\pi(R_1)^2$

$= 4\pi(\sqrt{3}k)^2$

$= 4\pi \times 3k^2$

$= 12\pi k^2$

Total surface area of hemisphere:

$\text{TSA}_{\text{hemisphere}} = 3\pi(R_2)^2$

$= 3\pi(2k)^2$

$= 3\pi \times 4k^2$

$= 12\pi k^2$

The ratio of total surface areas:

$\text{TSA}_{\text{sphere}} : \text{TSA}_{\text{hemisphere}} = 12\pi k^2 : 12\pi k^2$

$= 1 : 1$

Therefore, the ratio of their total surface areas is $1:1$ .

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If the ratio of the radii of the sphere and hemisphere is √3:2, then determine the ratio of their total surface area?

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