Linear Density Calculator

Calculate linear density, total mass, or length with metric and imperial units plus automatic tex, decitex, denier, g/m, kg/m, and lb/ft conversions.

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Linear Density Formula

• μ = linear density (kg/m)
• M = mass (kg)
• L = length (m)

For a homogeneous solid cylinder of constant diameter d and volumetric mass density ρ, linear density is:

In wave mechanics, linear density governs wave propagation speed on a stretched string under tension T:

Linear Density Definition

Linear density is mass per unit length, symbolized μ (mu) in physics. The SI unit is kg/m. It is useful for long objects such as wires, strings, ropes, textile fibers, and structural rods. It should be distinguished from linear charge density (C/m) and crystallographic linear density (atom centers per unit length).

Reference: Linear Density of Common Objects

Wire and rod values below are illustrative calculations using μ = ρπd²/4 and the stated material-density assumptions; actual products vary. Guitar-string values are rounded examples from OpenStax, not gauge specifications. The named carbon-fiber example uses the Toray T300 data sheet.

Object	Diameter	Linear Density	Notes
Guitar string, high E example	Not specified	3.1 x 10^-4 kg/m	Rounded OpenStax example
Guitar string, low E example	Not specified	5.8 x 10^-3 kg/m	Rounded OpenStax example
Piano wire, steel, 1 mm dia	1 mm	6.2 x 10^-3 kg/m (6.2 g/m)	Solid, rho = 7850 kg/m^3
Copper wire, AWG 10	2.59 mm	47.2 g/m	Heavy appliances, rho = 8960 kg/m^3
Copper wire, AWG 18	1.02 mm	7.4 g/m	Extension cords
Copper wire, AWG 24	0.511 mm	1.84 g/m	Signal wiring
Steel rod, 10 mm dia	10 mm	617 g/m	Rho = 7850 kg/m^3; approximate for 10 mm rebar
Carbon fiber tow, Toray T300 3K	7 μm individual filament	0.198 g/m (198 tex)	3,000 filaments; manufacturer reference average, not a bundle-diameter specification

Textile Fiber Units

Textile linear density describes mass per length without requiring a circular cross-section or a single material density. It can describe an individual fiber, a filament bundle, or a finished yarn; those are different objects. Common direct-count units are:

Unit	Definition	SI Equivalent	Primary Use
Denier (D)	grams per 9,000 m	1 D = 1.111 x 10^-7 kg/m	Yarn and filament size
Tex	grams per 1,000 m	1 tex = 1.0 x 10^-6 kg/m	ISO standard
Decitex (dtex)	grams per 10,000 m	1 dtex = 1.0 x 10^-7 kg/m	One tenth of a tex

Conversion: 1 tex = 9 denier = 10 dtex. Compare counts for the same type of object: a single fiber, a multifilament bundle, and a finished yarn can have very different linear densities. Use the specified product count or measure mass and length rather than treating a material name as a fixed denier value.

Crystallographic Linear Density

In materials science, crystallographic linear density counts atom centers on a specified direction per unit length. For ideal cubic lattices, the greatest linear atomic density occurs along <111> directions in BCC and <110> directions in FCC. Atomic packing helps describe slip systems, but it alone does not determine ductility or yield strength. This calculator computes mass-based linear density only.

Wave Speed and Guitar Strings

The two OpenStax string examples differ in linear density by about 19:1. In the ideal stretched-string model, f = v/(2L) and v = sqrt(T/μ), so lower μ gives a higher fundamental frequency at the same tension T and vibrating length L. Actual string sets can use different tensions and constructions. Winding adds mass while retaining a flexible core; its effect on stiffness depends on the construction.

Is linear density constant?

Linear density is constant when mass is uniformly distributed along the length. A uniform wire, rod, or uniformly wound section can have constant μ. A tapered shaft, a spliced string, or a beam whose section changes along its length can have position-dependent μ(x); M/L is the average over the measured length.

Does tension affect linear density?

If tension stretches an object while its mass stays fixed, μ decreases because length increases. For an elongation fraction ε, μ_new = μ_original/(1 + ε); a 0.1% elongation reduces μ by about 0.1%. Whether this matters depends on the material, load, and required accuracy. Record the measurement tension when comparing flexible or elastic samples.

How to Measure Linear Density

FAQ