How this calculator works
A sphere volume follows from its radius cubed. Diameter mode halves the measurement first. For a concentric hollow sphere, the inner volume is removed before dividing mass by volume. Material density describes the wall substance; bulk density uses the entire outer envelope, including the empty centre. Size mode reverses the calculation with a cube root and treats the given inner radius as fixed. The model assumes a spherical exterior, a concentric spherical cavity and uniform wall material. It excludes coatings, trapped contents and irregular voids.
Formula
Vmaterial = (4 pi/3)(R³ - r³); Vouter = (4 pi/3)R³; rho = m/Vmaterial; bulk density = m/Vouter; m = rho Vmaterial; R = cube root[3m/(4 pi rho) + r³]; diameter = 2R. For a solid sphere r = 0.
Worked example
A solid ball with radius 1 cm and mass 11.30973355 g occupies 4.188790205 mL. Its density is 2.7 g/cm³. A shell of the same material with outer radius 2 cm and inner radius 1 cm has mass 79.16813487 g. Its bulk density is only 2.3625 g/cm³ because one eighth of the outer volume is empty.
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Frequently asked questions
Should I enter radius or diameter?
Choose the measurement you actually took. Diameter crosses the full ball through its centre. Confusing the two changes calculated volume by a factor of eight.
Does water displacement measure shell material volume?
A sealed hollow ball displaces its outer envelope volume. That gives bulk density. Material volume requires the cavity to be excluded, for example by using its inner and outer radii.
Why are material and bulk density equal for a solid sphere?
With no cavity, both calculations divide the same mass by the same volume. A cavity makes bulk density smaller even when the wall material is unchanged.
How much does a measurement error matter?
For a solid sphere, a small relative radius error produces about three times that relative volume error. Thin-shell calculations are especially sensitive to errors in the two radii.