Thermal Expansion CalculatorΔL = α L₀ ΔT
Most materials grow when heated. Linear expansion is ΔL = α L₀ ΔT, where α is the coefficient of thermal expansion. For volume, ΔV = β V₀ ΔT, with β ≈ 3α for solids. Choose a material, enter the original length or volume and the temperature change to see how much it expands, so you can size expansion gaps for bridges, pipes, rails and decking, or headspace for liquids.
Formula verified against OpenStax — University Physics, Volume 2 · HyperPhysics, Georgia State University
By Manzoor Ahmad · Pharmacist & calculator developerReviewed & updated
Material & temperatures
α values are in µm/(m·K), the change in micrometres per metre per kelvin.
Step-by-step working
What is thermal expansion?
Thermal expansion is the tendency of matter to change in length, area or volume when its temperature changes. Almost all solids, liquids and gases expand when heated and contract when cooled.
Engineers design around it with bridge expansion joints, rail gaps, pipe loops and bimetallic thermostats. Thermometers use it directly: liquid in a thin tube expands as it warms.
Quick answers
How much does steel expand per degree?
About 12 µm per metre per °C. A 10 m steel beam grows 1.2 mm for every 10 °C rise.
How do I calculate the expansion gap for a pipe or deck?
Multiply α, the length and the full temperature range. A 6 m PVC pipe going from 10 to 60 °C needs at least 52 × 10⁻⁶ × 6 × 50 = 15.6 mm.
How much does water expand when heated?
About 0.02% per °C near room temperature (β ≈ 207 × 10⁻⁶ /K), rising at higher temperatures. Below 4 °C water actually contracts as it warms.
Why do glass jars crack in hot water?
The outside expands faster than the inside, creating stress. Borosilicate glass (α = 3.3) is used in lab glassware because it expands about a third as much as normal glass. The heat involved comes from the specific heat calculator.
Thermal expansion formulas
Valid for modest temperature ranges where α is roughly constant. A temperature change of 1 K equals 1 °C.
| Symbol | Meaning | SI unit |
|---|---|---|
| ΔL | Change in length | m |
| α | Linear expansion coefficient | 1/K |
| L₀ | Original length | m |
| ΔT | Temperature change | K or °C |
| ΔV, V₀ | Volume change and original volume | m³ |
| β | Volumetric expansion coefficient | 1/K |
How to calculate thermal expansion
- Look up α for the material (in 10⁻⁶ per K).
- Find the temperature change ΔT = T₂ − T₁.
- Multiply α × original length × ΔT.
- For volume, use β (≈ 3α for solids, or a measured β for liquids).
Worked example: a 100 m steel bridge
A 100 m steel bridge span (α = 12 × 10⁻⁶ /K) warms from 0 °C to 40 °C.
Over a full −20 °C to 45 °C range that grows to 78 mm, which is why bridges have expansion joints. A litre of water heated by 60 K expands by about 12.4 mL.
Coefficients of linear expansion
Near room temperature, in 10⁻⁶ per kelvin (µm per metre per °C).
| Material | α (×10⁻⁶ /K) | Growth of 1 m over 50 °C |
|---|---|---|
| Invar | 1.2 | 0.06 mm |
| Borosilicate glass | 3.3 | 0.17 mm |
| Soda-lime glass | 9 | 0.45 mm |
| Steel / concrete | 12 | 0.60 mm |
| Copper | 16.6 | 0.83 mm |
| Aluminium | 23.1 | 1.16 mm |
| PVC | 52 | 2.6 mm |
| HDPE | ≈ 200 | ≈ 10 mm |
Common mistakes to avoid
- Using α for volume. Volume expansion needs β ≈ 3α.
- Converting ΔT with offsets. A 10 °C change is a 10 K change; only Fahrenheit needs scaling (× 5/9).
- Ignoring restraint. If a part can’t move, the expansion turns into large thermal stress instead.
Manzoor Ahmad
Pharmacist turned Web Developer & Calculator Engineer · 13+ years in web developmentI’m a pharmacist by training who turned into a full-time tech geek, and for 13+ years I’ve designed and coded web tools. My pharmacy degree included physical pharmacy and pharmaceutics — states of matter, heat, flow and reaction kinetics — so the physics behind these tools is familiar ground, not a new hobby.
I’m not a physicist, so I don’t ask you to take my word for it: every formula on this page is traced to an open university textbook or NIST reference data, and the calculator is tested against published worked examples before it goes live. Spot an error? Tell me and it will be fixed and dated on the next update.
Thermal Expansion Calculator: frequently asked questions
Linear: ΔL = αL₀ΔT. Volume: ΔV = βV₀ΔT, with β ≈ 3α for isotropic solids.
The fractional change in length (or volume) per degree of temperature change, usually quoted in 10⁻⁶ per K.
Atoms vibrate more strongly, and because the forces between them are asymmetric, their average spacing increases.
Invar (an iron–nickel alloy) and fused silica have α near 1 × 10⁻⁶ /K or less, which is why they are used in precision instruments.
Yes, by about 9%. That is anomalous behaviour, unrelated to normal thermal expansion, and it is why pipes burst in winter.
References & sources
- OpenStax — University Physics, Volume 2 — Ch. 1.3, Thermal Expansion
- HyperPhysics, Georgia State University — thermal expansion coefficients
- OpenStax — College Physics 2e — Ch. 13.2, Thermal Expansion of Solids and Liquids
Disclaimer: This calculator is an educational tool built on idealised textbook models. It is not a substitute for professional engineering or safety calculations.