Example 1
α = 1.2×10⁻⁵, l₀ = 1 m, ΔT = 50 → Δl = 6×10⁻⁴ m.
Linear Expansion Calculator
Enter the coefficient α, the starting length, and the temperature rise. You get how much the bar grows. Steel at 12e-6 on one meter at 50 K is 0.6 mm. The same 50 K on 100 m is 6 cm.
Volume: ΔV = γ·V₀·ΔT (for solids γ ≈ 3α). Heat at ΔT: Q = c·m·ΔT.
Result
α (1/K), Length l₀ (m) and ΔT (K). The result shows up here.
How it works
When you heat a bar it grows along its length: Δl = α·l₀·ΔT. The final length is l = l₀ + Δl. For steel α ≈ 12e-6 1/K. One meter and 50 K give Δl = 0.0006 m, or 0.6 mm. Aluminum is often about 23e-6: two meters and 50 K is already 2.3 mm.
ΔT is a difference, not the number on a thermometer. Twenty Celsius to seventy is 50 K. Do not add 273. A negative ΔT means contraction: the same steel meter cooled by 50 K shrinks by 0.6 mm.
Type α in 1/K, from a material table. l₀ is in meters (or feet with US). Δl comes back in meters. On a 10 m rail and ΔT = 30 K, steel grows 3.6 mm. On 100 m and 50 K it grows 6 cm. A 1 cm gap on that rail is not enough.
For a solid the volume coefficient is about three times larger: γ ≈ 3α. Steel at 12e-6 linearly is about 36e-6 by volume. That factor of three lives on volumetric expansion.
This formula does not say how many joules you must add. Energy for the same ΔT is Q = c m ΔT. Here you only get geometry: how much longer the bar becomes.
Type α, l₀ and ΔT, then Calculate. The calculator example, 0.000012, 1 m and 50, should come back as 0.0006 m. A comma in 0.000012 parses the same as a period.
How to use
- Type α from a table, for example 0.000012 for steel, or 1.2e-5.
- Type the starting length l₀ in meters, for example 1, or 100 on a rail.
- Type ΔT as a rise: 50 when the temperature climbs 50 °C. Do not add 273.
- Click Calculate. 0.000012, 1 m and 50 K give 0.0006 m, or 0.6 mm.
- For volume with γ ≈ 3α, open volumetric expansion. For energy, open Q = c m ΔT.
Formula
Δl = α · l0 · ΔT
l = l0 + Δl. Unit of Δl: m.
Letters in Δl = α·l₀·ΔT
A bar grows along its length: Δl = α·l₀·ΔT. Steel at 12×10⁻⁶ on 1 m at 50 K is 0.6 mm; on 100 m the same gap is 6 cm.
- Δl
- Length increase in meters. 12e-6 × 1 × 50 = 0.0006 m, or 0.6 mm on one meter of steel.
- α
- Linear coefficient in 1/K. Steel is about 12×10⁻⁶; aluminum near 23×10⁻⁶ already gives 2.3 mm on 2 m at 50 K.
- l₀
- Starting length in meters. One hundred meters of rail at 50 K and steel is Δl = 6 cm, not 0.6 mm.
- ΔT
- Temperature jump in kelvin. 20 °C to 70 °C is 50 K; you do not add 273 to a difference.
Real-life examples
Example 2
α = 1.2×10⁻⁵, l₀ = 10 m, ΔT = 30.
Example 3
α ≈ 2.3×10⁻⁵, l₀ = 2 m.
Example 4
α ≈ 1.7×10⁻⁵, l₀ = 1 m.
Example 5
α ≈ 9×10⁻⁶, l₀ = 0.5 m.
Example 6
α ≈ 1.2×10⁻⁵, l₀ = 5 m.
Example 7
l₀ = 1 m, ΔT = −40: contraction.
Example 8
α = 1.2×10⁻⁵, l₀ = 0.2 m.
Example 9
l₀ = 50 m, steel.
Example 10
α ≈ 1.2×10⁻⁶, l₀ = 1 m, ΔT = 50.
Ways to use this calculator
- You check whether a 1 cm gap is enough on a 100 m rail for a 50 K jump.
- You compare steel at 12e-6 with aluminum at 23e-6 on the same meter.
Frequently asked questions
How much Δl for steel α = 12e-6, l₀ = 1 m and ΔT = 50 K?
Δl = 0.0006 m = 0.6 mm. One hundred kelvin on that meter is 1.2 mm.
Is 50 K the same as a 50 °C difference?
Yes. 20 °C to 70 °C is 50 K. You do not add 273 to a difference.
How much on a 100 m rail at α = 12e-6 and ΔT = 50 K?
Δl = 0.06 m = 6 cm. A 1 cm gap is too tight for that jump.
How much at 10 m, steel and ΔT = 30 K?
Δl = 0.0036 m = 3.6 mm. Final length is 10.0036 m after that rise.
What α should I type for aluminum?
On the order of 23e-6 1/K. Two meters and 50 K give Δl = 0.0023 m.
Can ΔT be negative?
Yes. Cooling a steel meter by 50 K shortens it by 0.6 mm.
How is γ different from α?
γ is volumetric. For solids γ ≈ 3α. Steel at 12e-6 linearly is about 36e-6 by volume.
Does this formula compute joules?
No. Energy for that ΔT is Q = c m ΔT. Here you only get the length change.
Does 1.2e-5 work instead of 0.000012?
Yes. 1.2e-5, 12e-6 and 0.000012 are the same steel α.
Does a comma in 0.000012 work?
Yes. 0,000012 and 0.000012 are the same coefficient.
Knowledge sources
The formula is the school one. Units follow SI; NIST SP 330 and BIPM define the measures, not your result.
Page updated in 2026.