Calculate ideal HCP unit-cell height c or sphere radius r from the other value in inches, feet, cm, or m. The model assumes identical touching spheres; real materials can differ.

Ideal HCP geometry: identical touching spheres, with height c spanning two layer spacings.

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HCP Height Formula

The ideal HCP height is calculated from the atom radius using the height-to-radius relationship for a hexagonal close-packed unit cell.

H = 4r√((2) / (3))
r = (H) / (4√((2) / (3)))
  • H = c = ideal HCP unit-cell height, spanning two adjacent-layer spacings
  • r = radius of an identical touching sphere / atom
  • \(\sqrt{2/3}\) = geometric factor for the ideal HCP structure

The calculator has two functions:

  • Calculate HCP height: Enter the radius, and the calculator applies H = 4r√(2/3).
  • Calculate radius: Enter the HCP height, and the calculator rearranges the formula to r = H ÷ [4√(2/3)].

All values are converted through meters internally using exact inch, foot and centimeter definitions, then converted to your selected result unit.

HCP Height and Radius Multipliers

For an ideal HCP structure, the HCP height is about 3.266 times the radius.

Quantity Multiplier Use
HCP height from radius ≈ 3.2660 × radius Use when radius is known
Radius from HCP height ≈ 0.3062 × HCP height Use when HCP height is known

Supported Length Units

Unit Abbreviation Inches Equivalent
Inch in 1 in
Foot ft 12 in
Centimeter cm ≈ 0.393701 in
Meter m ≈ 39.3701 in

Example Problems

Example 1: Find HCP height from radius

If the radius is 2 cm, use the formula:

H = 4(2)√((2) / (3))

The result is:

H ≈ 6.5320 cm

Example 2: Find radius from HCP height

If the HCP height is 10 in, use the rearranged formula:

r = (10) / (4√((2) / (3)))

The result is:

r ≈ 3.0619 in

FAQ

What does HCP height mean?

Here HCP height means the ideal unit-cell repeat height c in ABAB stacking, spanning two adjacent-layer spacings. The spacing between adjacent layers is c/2. Real HCP materials may have a different c/a ratio, so atomic radius alone does not predict their measured unit-cell height.

Why is the formula based on radius?

In an ideal HCP structure, atoms are modeled as touching spheres. The radius determines the spacing between atoms, so the unit cell height can be calculated from that radius using the geometric factor 4√(2/3).

Can I enter radius and HCP height in different units?

Yes. You can enter one value in inches, feet, centimeters, or meters and choose a different output unit for the missing value. The calculator handles the unit conversion automatically.