Checked example
Worked calculation
Result: At r = 5, the cylinder volume is 785.398163 and the sphere volume is 523.598776 cubic units.
01 • Guide
The enclosing dimensions
A snug cylinder around a sphere has the same radius and a height equal to the sphere’s diameter, 2r. Forgetting that height causes the comparison to be inconsistent.
Keep the full precision of intermediate values and round only the final display. This avoids compounding error when a derived radius or converted unit is used in another step.
02 • Guide
Archimedes’ ratio
Dividing sphere volume by enclosing-cylinder volume gives (4/3)/(2) = 2/3. The remaining one third belongs to the regions inside the cylinder but outside the sphere.
The calculator validates physical limits before displaying a result. Blank or impossible inputs never produce a stale value, NaN or Infinity.
03 • Guide
How scaling behaves
Both volumes scale with r³. Doubling radius multiplies each volume by eight but never changes the two-thirds ratio, as the live comparison graph demonstrates.
Interpretation
How to use the result responsibly
The answer models an ideal geometric cylinder. Manufacturing tolerances, wall thickness, rounded ends, deformation, tank tilt and measurement uncertainty can all shift a real value. Use inside dimensions for capacity and outside dimensions for occupied space. For safety, compliance, dosing, structural or purchasing decisions, compare the estimate with drawings, calibration data or a qualified professional.
Dimensional checks help catch mistakes: a volume result must carry cubic units, an area result square units and a derived radius or height linear units. If changing one unit alters the represented physical size, a conversion step was missed.