Sphere Density Calculator
Two tools in one. Mode 1 computes the bulk (material) density of a single solid sphere from its mass and radius — ρ = m / V, with V = 4/3·π·r³. Mode 2 computes sphere packing density: the packing fraction of SC, BCC, FCC, HCP or random arrangements, the maximum number of balls that fit in a container, and the bulk density of the packed bed.
Mode 1 — Density of a solid sphere
Mode 2 — Packing density & ball count
Results
Enter values and press Calculate.
Formulas used
Vsphere = (4/3)·π·r³ · ρ = m / V (mode 1)η = packing fraction · N = η·Vcontainer / Vsphere · ρbulk = η·ρmaterial (mode 2)
Worked example
A steel ball weighs 4.2 kg and has radius 0.1 m.
V = 4/3·π·(0.1)³ = 0.00418879 m³ → ρ = 4.2 / 0.00418879 = 1002.68 kg/m³ (≈ 1.003 g/cm³).
Packing the same steel (ρ = 7800 kg/m³) as FCC spheres of r = 0.05 m into a 1 m³ container: each sphere is 5.236×10⁻⁴ m³, so N = 0.74048 / 5.236×10⁻⁴ ≈ 1414 balls, and the packed bed's bulk density is 0.74048 × 7800 = 5775.74 kg/m³.
Packing fraction reference
| Arrangement | Symbol | Packing fraction η | Use when |
|---|---|---|---|
| Face-centered cubic | FCC | π/(3√2) ≈ 0.74048 | Deliberately stacked layers (densest possible) |
| Hexagonal close packing | HCP | π/(3√2) ≈ 0.74048 | Deliberately stacked layers (densest possible) |
| Body-centered cubic | BCC | √3·π/8 ≈ 0.68017 | Ordered lattice, e.g. crystal models |
| Simple cubic | SC | π/6 ≈ 0.52360 | Grid-stacked spheres |
| Random close packing | RCP | ≈ 0.64 | Balls poured / shaken into a container |
| Random loose packing | RLP | ≈ 0.55 | Balls gently placed, no settling |
Scope boundary
Mode 1 answers “what is the material density of this ball?” — one solid sphere, mass ÷ volume. Mode 2 answers “how do spheres pack?” — packing fraction, ball count and bulk density of the packed bed including air gaps. The two densities differ: steel is 7800 kg/m³ as a material, but a container of packed steel balls is only ~5776 kg/m³ in FCC arrangement.
Related tools: density unit converter (convert kg/m³ ↔ g/cm³ ↔ lb/ft³). This tool does not compute surface area or solve sphere equations — see the sphere geometry tools for those.
Frequently asked questions
What is the density of a sphere?
The density of a solid sphere is its mass divided by its volume, ρ = m / V, where V = (4/3)·π·r³. Measure or weigh the sphere, cube its radius, multiply by 4.18879, then divide the mass by that volume. The calculator above does this in one step and converts the result into kg/m³, g/cm³, lb/ft³ and oz/in³.
What is the formula for the volume of a sphere?
The volume of a sphere is V = (4/3)·π·r³ ≈ 4.18879·r³. If you know the diameter instead of the radius, use r = d/2, giving V = π·d³/6 ≈ 0.52360·d³. The calculator accepts either radius or diameter in mm, cm, m, in, ft or yd.
What is sphere packing density?
Sphere packing density (packing fraction η) is the fraction of a container's volume actually occupied by spheres: η = (number of spheres × sphere volume) / container volume. The rest is empty void space. For identical hard spheres, the densest possible ordered arrangements (FCC and HCP) reach η ≈ 0.74048, meaning about 26% of the space always stays empty.
What are the packing fractions for SC, BCC, FCC and HCP?
For identical spheres: simple cubic (SC) packs at π/6 ≈ 0.52360, body-centered cubic (BCC) at √3·π/8 ≈ 0.68017, and both face-centered cubic (FCC) and hexagonal close packing (HCP) at π/(3√2) ≈ 0.74048 — the Kepler limit, the densest any ordered arrangement can reach. Random close packing of poured spheres reaches about 0.64 and random loose packing about 0.55.
How do I estimate how many balls fit in a container?
Use mode 2 of this calculator: pick the packing arrangement (for poured balls use RCP ≈ 0.64), enter the container volume and the ball radius, and it computes N = η·V_container / V_sphere. This is the theoretical upper bound — real packing near walls is slightly lower because of boundary effects.
What is bulk density of a pile of spheres?
Bulk density is the mass of a packed bed of spheres divided by the total volume it occupies, including the air gaps: ρ_bulk = η·ρ_material, where η is the packing fraction and ρ_material is the density of the solid sphere material. For example, steel balls (7800 kg/m³) in FCC packing (η = 0.74048) have a bulk density of about 5776 kg/m³.
Does a sphere's density change with its size?
No — density is an intensive property of the material, so a bigger solid sphere of the same material has exactly the same density. What changes with size is the total mass and the volume, both of which grow with the cube of the radius (V = 4/3·π·r³), so they cancel out in the ratio m/V.
How do I find the density of a ball without knowing its material?
Weigh the ball on a scale to get its mass, measure its diameter with calipers (divide by 2 for the radius), then compute ρ = m / (4/3·π·r³). Compare the result against known material densities — for instance, aluminum is about 2700 kg/m³ (2.70 g/cm³) and steel about 7800 kg/m³ (7.80 g/cm³).
What is the difference between bulk density and material density?
Material density is mass per unit volume of the solid substance itself (no air gaps). Bulk density is mass per unit volume of a pile or container of objects, including the void space between them. Mode 1 of this tool computes material density of a single sphere; mode 2 computes bulk density of a packed bed from the packing fraction.
Why can't spheres fill 100% of a container?
Identical spheres always leave gaps: even the densest ordered stacking (FCC/HCP, proven optimal by the Kepler conjecture, resolved in 2014) fills only about 74.05% of space, leaving roughly 26% empty. Randomly poured spheres do worse, around 64%. No arrangement of equal spheres can exceed the π/(3√2) ≈ 0.74048 limit.
How do I convert sphere density between kg/m³ and g/cm³?
Divide kg/m³ by 1000 to get g/cm³ — so 7800 kg/m³ (steel) equals 7.8 g/cm³. The calculator shows every result in kg/m³, g/cm³, kg/L, lb/ft³ and oz/in³ at once, so no manual conversion is needed.
What packing fraction should I use for randomly poured balls?
Use random close packing, η ≈ 0.64, for balls that were poured or shaken into a container and settled. Use random loose packing, η ≈ 0.55, for balls gently placed with no settling. Ordered crystalline arrangements (SC, BCC, FCC, HCP) only apply when spheres are deliberately stacked in a lattice, as in ball bearings or crystal structures.
Is this sphere density calculator accurate?
The physics is exact: sphere volume uses V = 4/3·π·r³ with full floating-point π, ordered-lattice packing fractions use their closed-form values (π/6, √3·π/8, π/(3√2)), and unit conversions use exact NIST definitions. The sphere-count estimate N = η·V_container/V_sphere is an ideal upper bound — real counts near container walls run a few percent lower due to boundary effects.
Can I use this for shipping or storage planning?
Yes, as a first estimate: enter your container volume, ball size and material density in mode 2 to get the packed mass (ρ_bulk × container volume) and the theoretical maximum ball count. For logistics quotes, add a 5–10% safety margin below the computed count, because wall effects and imperfect settling reduce real packing below the ideal fraction.
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