m
Please enter original length.
×10⁻⁶
Please enter coefficient α.
Steel: 12 • Aluminum: 23.1 • Copper: 17 • Concrete: 12 • Glass: 8.5
°C
Enter initial temperature.
°C
Enter final temperature.
Change in Length (ΔL)
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Sources & Methodology
Calculations use standard thermodynamics formulas for linear and volumetric thermal expansion, verified against Engineering Toolbox material data and NIST physical constants.
1
Engineering Toolbox — Coefficients of Linear Thermal Expansion
Reference database for linear thermal expansion coefficients (α) for metals, polymers, building materials, and liquids used in all material presets.
2
NIST — Physical Constants and Unit Definitions
SI unit definitions for length (meter), temperature (Kelvin), and the relationship β ≈ 3α for isotropic solids used in volumetric calculations.
3
Khan Academy — Thermal Expansion
Educational reference for thermal expansion formulas, physical interpretation, and engineering applications used in the content section.
Methodology: Linear expansion: ΔL = α × L₀ × ΔT; Final length L = L₀ + ΔL = L₀(1 + αΔT). Volumetric expansion: ΔV = β × V₀ × ΔT; Final volume V = V₀(1 + βΔT). Coefficients entered as ×10⁻⁶/°C. ΔT = T₂ − T₁.
⏱ Last reviewed: March 2026
How to Calculate Thermal Expansion — Linear and Volumetric
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature. Almost all materials expand when heated and contract when cooled. Understanding thermal expansion is critical in engineering, construction, manufacturing, and materials science.
Linear Thermal Expansion Formula
ΔL = α × L₀ × ΔT
ΔL = Change in length (meters)
α = Coefficient of linear thermal expansion (per °C or per K)
L₀ = Original length (meters)
ΔT = Temperature change = T₂ − T₁ (°C)
Example: A 10 m steel beam (α = 12×10⁻⁶/°C) heated by 50°C:
ΔL = 12×10⁻⁶ × 10 × 50 = 0.006 m = 6 mm expansion
α = Coefficient of linear thermal expansion (per °C or per K)
L₀ = Original length (meters)
ΔT = Temperature change = T₂ − T₁ (°C)
Example: A 10 m steel beam (α = 12×10⁻⁶/°C) heated by 50°C:
ΔL = 12×10⁻⁶ × 10 × 50 = 0.006 m = 6 mm expansion
Volumetric Thermal Expansion Formula
ΔV = β × V₀ × ΔT
β = Coefficient of volumetric thermal expansion ≈ 3α (for isotropic solids)
V₀ = Original volume (m³)
Example: 1 m³ of water (β = 207×10⁻⁶/°C) heated by 30°C:
ΔV = 207×10⁻⁶ × 1 × 30 = 0.00621 m³ = 6.21 liters expansion
V₀ = Original volume (m³)
Example: 1 m³ of water (β = 207×10⁻⁶/°C) heated by 30°C:
ΔV = 207×10⁻⁶ × 1 × 30 = 0.00621 m³ = 6.21 liters expansion
Thermal Expansion Coefficients of Common Materials
| Material | α (×10⁻⁶/°C) | β (×10⁻⁶/°C) | Common Application |
|---|---|---|---|
| Aluminum | 23.1 | 69.3 | Aircraft, window frames |
| Brass | 19.0 | 57.0 | Valves, fittings |
| Copper | 17.0 | 51.0 | Electrical wiring, pipes |
| Steel (carbon) | 12.0 | 36.0 | Bridges, buildings |
| Concrete | 12.0 | 36.0 | Roads, structures |
| Iron (cast) | 10.8 | 32.4 | Pipes, engine blocks |
| Glass (ordinary) | 8.5 | 25.5 | Windows, containers |
| Fused silica | 0.55 | 1.65 | Precision optics |
| Invar | 1.2 | 3.6 | Precision instruments |
| Water (25°C) | — | 207 | Thermal storage, cooling |
Why Thermal Expansion Matters in Engineering
💡 Bridge expansion joints: The Golden Gate Bridge (2,737 m total length, steel) expands and contracts by approximately 2,737 × 12×10⁻⁶ × 50°C = 1.64 m between the coldest winter night and hottest summer day. Expansion joints accommodate this movement safely.
Real-World Thermal Expansion Applications
- Railroad tracks: Gaps are left between rail sections to allow expansion. A 1 km steel track expands about 12 mm per 10°C temperature rise.
- Concrete roads and sidewalks: Expansion joints every 5–6 meters prevent buckling during summer heat. Without them, concrete can heave dramatically.
- Bimetallic strips in thermostats: Two metals with different α values bond together and bend when heated, mechanically triggering electrical contacts.
- Engine pistons: Aluminum pistons (α = 23.1) are machined slightly smaller than the cylinder bore to account for thermal expansion when the engine warms up.
- Pipe systems: Hot water pipes use expansion loops, bellows, or flexible couplings because a 50-meter copper pipe heated by 60°C expands 51 mm.
- Glass-ceramic cookware: Low-expansion ceramics (α ≈ 1×10⁻⁶/°C) resist thermal shock, while ordinary glass can shatter from rapid temperature changes.
Thermal Contraction — Cooling and Negative ΔT
Thermal expansion works in reverse when materials cool. A negative ΔT (final temperature lower than initial) gives a negative ΔL or ΔV, meaning the material has shrunk. This is called thermal contraction. The same formula applies — shrink-fitting of metal parts uses controlled cooling to temporarily contract a part so it slides into a housing, then expands to form a tight interference fit.
Frequently Asked Questions
The linear thermal expansion formula is ΔL = α × L₀ × ΔT, where ΔL is the change in length, α is the coefficient of linear thermal expansion in 1/°C, L₀ is the original length, and ΔT is the temperature change. The final length is L = L₀(1 + αΔT).
Steel has α = 12 × 10⁻⁶/°C. Formula: ΔL = 12×10⁻⁶ × L₀ × ΔT. For a 10 m steel beam heated 50°C: ΔL = 12×10⁻⁶ × 10 × 50 = 0.006 m = 6 mm. This is why steel bridges use expansion joints — seasonal temperature swings of 50°C cause significant length changes that could buckle the structure if unconstrained.
Linear expansion (ΔL = αL₀ΔT) measures one-dimensional length change and applies to beams, rods, and pipes. Volumetric expansion (ΔV = βV₀ΔT) measures three-dimensional volume change. For isotropic materials (same properties in all directions), β ≈ 3α. Volumetric expansion is used for liquids, gases, and bulk solids.
Among common engineering materials, aluminum has a relatively high linear coefficient of α = 23.1 × 10⁻⁶/°C. Polymers and plastics expand even more — polyethylene has α around 200 × 10⁻⁶/°C. Liquids have much higher volumetric coefficients; water is 207 × 10⁻⁶/°C and ethanol is about 1,100 × 10⁻⁶/°C.
Steel and concrete expand significantly with temperature. A 100 m steel bridge spans experiences about 60 mm of length change over a 50°C seasonal temperature range (ΔL = 12×10⁻⁶ × 100 × 50 = 0.06 m). Without expansion joints, this force would buckle the deck, crack abutments, or damage bearings. Expansion joints allow the bridge to grow and shrink freely while maintaining structural integrity.
Use ΔL = α × L₀ × ΔT. For a 5 m copper pipe (α = 17×10⁻⁶) heated from 20°C to 80°C: ΔL = 17×10⁻⁶ × 5 × 60 = 0.0051 m = 5.1 mm. Plumbing codes require expansion loops, offsets, or flexible connectors in hot water systems to absorb this movement and prevent joint stress.
Concrete has a coefficient of linear thermal expansion of approximately 10–12 × 10⁻⁶/°C, very similar to steel (12 × 10⁻⁶). This compatibility is one of the main reasons steel reinforcement works so well in concrete — both materials expand and contract at nearly the same rate, preventing internal stress cracking at the steel-concrete interface.
A negative ΔL or ΔV means the material is contracting — the final temperature is lower than the initial, so ΔT is negative. This is thermal contraction. The magnitude tells you how much smaller the material has become. Engineers use controlled cooling (shrink fitting) to temporarily contract metal parts for assembly, then allow them to warm up and expand into a tight fit.
A 1 km steel rail expands by ΔL = 12×10⁻⁶ × 1000 × 10 = 0.12 m = 12 cm for every 10°C temperature rise. Traditional tracks left small gaps between rails to accommodate this. Modern continuous welded rail (CWR) is installed under tension at a specific temperature so expansion and contraction are balanced, but the rail is anchored to prevent buckling called "sun kinking."
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