What is the volume of a cube if the diagonal of one of the sides is $\sqrt{50}$cm?

Analyze the question: The "diagonal of one of the sides" is a face diagonal of the cube. We need the edge length first, then the volume.

Concept: For a square face with side a, the diagonal d is given by Pythagoras: d² = a² + a² = 2a²

Solution:

1. Given d = √50, so d² = 50.
2. 2a² = 50, so a² = 25, so a = 5 cm.
3. Volume of a cube = a³ = 5³ = 125 cm³.

Check: A face with side 5 cm has diagonal √(25 + 25) = √50 ✓.

Answer: The volume is 125.0 cm³, which is option C.

Tip: For a face diagonal, a = d/√2. Here √50/√2 = √25 = 5. Do not confuse the face diagonal (a√2) with the space diagonal (a√3). Always check which one the question gives you.