Cylinder height from sphere volume equality: A circular cylinder has the same radius as a sphere and their volumes are equal. The height of the cylinder is equal to what multiple of its radius?

Introduction / Context:
Equating cylinder and sphere volumes with common radius yields a direct relationship between cylinder height and radius. This is a standard mensuration identity.

Given Data / Assumptions:

• Common radius r.
• V_cylinder = π r^2 h; V_sphere = (4/3) π r^3; volumes equal.

Concept / Approach:

• Set π r^2 h = (4/3) π r^3 and solve for h in terms of r.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 2/3 r or r: Too short to produce equal volume.
• Diameter (2r): Too tall; volume would exceed the sphere volume.

Common Pitfalls:

• Dropping the 1/3 in sphere’s volume.
• Confusing diameter and radius when interpreting options.

Final Answer:

4/3 times its radius