CUET UG 2025 — General Test Geometry

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

Let the radius of sphere $R_1 = \sqrt{3}k$R1​=3​k and radius of hemisphere $R_2 = 2k$R2​=2k, where $k$k is a constant.

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The total surface area of a sphere is $4\pi r^2$4πr2.

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$2πr2+πr2=3πr2.

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Total surface area of sphere:

$$\text{TSA}_{\text{sphere}} = 4\pi(R_1)^2$$TSAsphere​=4π(R1​)2

$$= 4\pi(\sqrt{3}k)^2$$=4π(3​k)2

$$= 4\pi \times 3k^2$$=4π×3k2

$$= 12\pi k^2$$=12πk2

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Total surface area of hemisphere:

$$\text{TSA}_{\text{hemisphere}} = 3\pi(R_2)^2$$TSAhemisphere​=3π(R2​)2

$$= 3\pi(2k)^2$$=3π(2k)2

$$= 3\pi \times 4k^2$$=3π×4k2

$$= 12\pi k^2$$=12πk2

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The ratio of total surface areas:

$$\text{TSA}_{\text{sphere}} : \text{TSA}_{\text{hemisphere}} = 12\pi k^2 : 12\pi k^2$$TSAsphere​:TSAhemisphere​=12πk2:12πk2

$$= 1 : 1$$=1:1

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