Surface area ratio from cube volume ratio: Two cubes have volumes in the ratio 1 : 27. Find the ratio of their surface areas.

Introduction / Context:
For similar solids, volumes scale as the cube of the linear scale factor, and surface areas scale as its square. Extract the edge ratio first from the volume ratio, then square for surface areas.

Given Data / Assumptions:

• V1 : V2 = 1 : 27
• Edge ratio s1 : s2 = ∛(1) : ∛(27) = 1 : 3

Concept / Approach:
Surface area ratio = (s1 : s2)^2 = 1^2 : 3^2 = 1 : 9.

Step-by-Step Solution:
Edge ratio = 1 : 3Surface area ratio = 1^2 : 3^2 = 1 : 9

Verification / Alternative check:
Pick sample edges 1 and 3; areas are 6*1^2 and 6*3^2 → 6 : 54 → 1 : 9 after canceling 6.

Why Other Options Are Wrong:
1:3 confuses linear with area; 1:8 belongs to cubes vs squares; 1:18 mixes factors; 1:27 is the volume ratio, not area.

Common Pitfalls:
Forgetting to take cube root for edge ratio before squaring for area.

Final Answer:
1 : 9