Cubic Cell Calculator (SC, BCC, FCC: edge length, atomic radius, density)

Understanding the cubic unit cell calculator — concept and principle

Many metals crystallise in simple cubic, body-centred cubic (BCC) or face-centred cubic (FCC) lattices. This calculator converts between edge length and atomic radius, and gives atoms per cell, coordination number, packing efficiency and density.

Four atoms in a cube 352.4 pm on a side give nickel a density of 8.908 g/cm³.

How it is calculated

Simple cubic: a = 2r; BCC: a = 4r ÷ √3; FCC: a = 2√2 r.
Atoms per cell Z: 1 (SC), 2 (BCC), 4 (FCC); coordination numbers 6, 8, 12; packing efficiencies about 52%, 68%, 74%.
Density = Z × M ÷ (NA × a³), with a in cm (1 pm = 10⁻¹⁰ cm) and NA = 6.02214076 × 10²³ mol⁻¹.
Unit-cell volume in Å³ = (a in pm ÷ 100)³.

The principle

Atoms are modelled as hard spheres that touch along a particular direction: along the edge in SC, along the body diagonal in BCC and along the face diagonal in FCC. That fixes the relation between a and r. Counting shared positions (a corner atom belongs ⅛ to each cell, a face atom ½) gives atoms per cell. Knowing how many atoms of known mass fit in a known volume gives the density, which is a quick check on which lattice a metal adopts.

Worked example

Default inputs: FCC, edge length 352.4 pm, molar mass 58.693 g/mol (nickel).

r = 352.4 ÷ (2√2) ≈ 124.59 pm.
Atoms per cell = 4; coordination number = 12; packing efficiency = 74%.
Cell volume = (3.524 Å)³ ≈ 43.763 Å³ = 4.3763 × 10⁻²³ cm³.
Density = 4 × 58.693 ÷ (6.02214 × 10²³ × 4.3763 × 10⁻²³) ≈ 8.908 g/cm³.
Atoms per cm³ = 4 ÷ 4.3763 × 10⁻²³ ≈ 9.14 × 10²².

Each point is the calculator re-run with only “Length (pm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Length (pm)	Density (g/cm³)
88.1	570.122
176	71.265
264	21.116
352	8.908
440	4.561
529	2.639
705	1.114

How to read the result

The computed 8.908 g/cm³ matches nickel's measured density closely, confirming its FCC structure. If a computed density is far from the measured value, the assumed lattice or edge length is probably wrong. FCC and hexagonal close packing share the highest packing efficiency for equal spheres, 74%.

Common mistakes

Using the wrong touching direction for the lattice.
Forgetting to convert pm to cm before cubing.
Using Z = 1 for BCC or FCC.
Treating atomic radius as a fixed property; it depends on bonding and coordination.

Keep in mind: The hard-sphere model is an idealisation. Use measured crystallographic data for research.

Frequently asked questions

How many atoms are in an FCC unit cell?

Four: 8 corners × ⅛ + 6 faces × ½.

What is the packing efficiency of BCC?

About 68%.

How do you calculate density from a unit cell?

Density = atoms per cell × molar mass ÷ (Avogadro's number × cell volume).

Sources: Callister, W. D. & Rethwisch, D. G., Materials Science and Engineering: An Introduction (crystal structures) · Kittel, C., Introduction to Solid State Physics, 8th ed. Wiley

Checked 9 Nov 2026. OpenStax, Chemistry 2e (2019), §10.6 “Lattice Structures in Crystalline Solids” (SC/BCC/FCC atoms per cell, coordination numbers, packing, edge–radius relations, density examples for Po, Ni, Ca, Ag); Avogadro constant exact (2019 SI).

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