Large sphere split into 64 equal small spheres — find each small sphere’s surface area: A big sphere of diameter 8 cm is divided into 64 identical small spheres. What is the surface area of one small sphere?

Introduction / Context:
Equal-volume partition implies that the volume of the large sphere equals 64 times the volume of each small sphere. Use this to derive each small sphere’s radius, then compute its surface area.

Given Data / Assumptions:

• Large diameter = 8 cm ⇒ large radius R = 4 cm.
• Number of small spheres = 64.
• Volume conservation: (4/3)πR^3 = 64 * (4/3)πr^3.

Concept / Approach:
Cancel constants and solve for r. Then S_small = 4πr^2.

Step-by-Step Solution:
R^3 = 64 r^3 ⇒ r^3 = R^3 / 64 = 64 / 64 = 1 ⇒ r = 1 cmS_small = 4πr^2 = 4π * 1^2 = 4π cm2

Verification / Alternative check:
Total surface area is not conserved—only volume is. The question asks for one small sphere, which is 4π cm2.

Why Other Options Are Wrong:
π and 2π are too small; 8π is double the correct area.

Common Pitfalls:
Assuming surface area is conserved during partitioning; it is not.

Final Answer:
4 π cm2