The areas of two spheres are in the ratio 1 : 4. The ratio of their volumes is-

Question: The areas of two spheres are in the ratio 1 : 4. The ratio of their volumes is-

Solution:
Let the radii of the two spheres be r₁ and r₂.
Surface area of a sphere = 4πr²

According to the question,
(4πr₁²)/(4πr₂²) = 1/4
⇒ r₁²/r₂² = 1/4
⇒ r₁/r₂ = √(1/4) = 1/2

Volume of a sphere = (4/3)πr³
∴ Volume ratio = {(4/3)πr₁³}/{(4/3)πr₂³}
= r₁³/r₂³
= (r₁/r₂)³
= (1/2)³
= 1/8

∴ The ratio of their volumes is 1 : 8.