Thermal Expansion Calculator: ΔL, ΔA, ΔV by Material
Calculate how materials change size with temperature using linear, area, and volume expansion formulas. Compare materials and analyze gap changes in multi-material assemblies.
If a buried pipeline has buckled into S-shaped waves between fixed anchors, you're looking at thermal expansion that had nowhere to go. Same diagnostic for railroad sun-kink failures, sagging power lines on a hot afternoon, and aluminum gutters that bow out by August. The math is short: ΔL = α L₀ ΔT for linear expansion. Aluminum's α is 23.1 μm/m/°C (NIST), so a 30 m run through a 40°C summer-to-winter swing grows or shrinks by 28 mm. Carbon steel sits at about 11.7 μm/m/°C, while Invar (Fe-Ni alloy) drops below 1, which is why Invar gets used in precision instruments. Skip the expansion allowance and something has to bend, buckle, or crack to absorb the displacement.
Quick Reference: Thermal Expansion Coefficients
| Material | α (×10⁻⁶/°C) | Temperature Range | Source |
|---|---|---|---|
| Aluminum 6061-T6 | 23.6 | 20 to 100°C | ASM Handbook Vol. 2 |
| Carbon Steel (AISI 1020) | 11.7 | 20 to 100°C | ASM Handbook Vol. 1 |
| Stainless Steel 304 | 17.3 | 20 to 100°C | ASM Handbook Vol. 1 |
| Copper (C11000) | 16.5 | 20 to 100°C | ASM Handbook Vol. 2 |
| Brass (C26000) | 19.9 | 20 to 100°C | ASM Handbook Vol. 2 |
| Invar 36 | 1.3 | 20 to 100°C | NIST SRD 100 |
| Borosilicate Glass | 3.3 | 20 to 300°C | Schott Technical Data |
| Fused Silica | 0.55 | 20 to 1000°C | Heraeus Technical Data |
| HDPE | 100 to 200 | 20 to 60°C | ASTM D696 |
| Concrete (typical) | 10 to 14 | 20 to 100°C | ACI 318 |
Values from ASM Handbooks, NIST Standard Reference Data, and manufacturer specifications. CTE varies with temperature and alloy composition; use material-specific data for critical applications. β (area) ≈ 2α, γ (volume) ≈ 3α for isotropic materials.
The Four Modes: Conduction, Convection, Radiation, Phase Change
Thermal expansion sits inside a wider thermal context. Heat moves into and out of a part by four distinct mechanisms, and which one dominates decides the temperature field that drives expansion. A solar-loaded steel rail picks up heat by radiation from the sun and convection from ambient air. A copper bus bar in a switchgear cabinet heats up by ohmic dissipation and sheds heat by free convection. A cryogenic LNG line cools by conduction through its insulation jacket and by latent boil-off at the inner surface.
Conduction
Fourier's law q = -k∇T. Aluminum k = 237 W/mK, copper k = 400 W/mK, 304 stainless k = 16.2 W/mK. A heated end of an aluminum bar reaches the cold end fast; the same geometry in stainless lags by an order of magnitude, and the temperature gradient across the part is what governs differential expansion stress.
Convection
Newton's law of cooling q = h(T_s - T_∞). Free convection in air gives h = 5 to 25 W/m²K. Forced air bumps it to 25 to 250. Water can hit 1000 to 20,000. The surface coefficient sets how quickly a part equilibrates with its environment, and that controls the timescale of expansion.
Radiation
Stefan-Boltzmann q = εσ(T⁴ - T_surr⁴). Polished aluminum ε ≈ 0.05, oxidized steel ε ≈ 0.8, lampblack ε ≈ 0.97. A roof in full sun can run 30°C above ambient even with breeze, and a bare steel deck plate that's 30°C hotter than its supports expands accordingly.
Phase Change
Latent heat dwarfs sensible heat. Water absorbs 2257 kJ/kg vaporizing at 100°C, the same energy as heating it from 0°C to 540°C if it stayed liquid. A steam line that condenses on a cold flange dumps a huge thermal load locally and creates a hotspot that expands faster than its neighbors.
For a bare expansion calculation you only need ΔT. But predicting ΔT from environmental conditions means tracking which modes dominate. A lot of failed expansion-joint designs trace back to assuming uniform body temperature when one face was actually radiating into a cold sky and the back face was glued to a warm wall.
Thermal Resistance Networks: Stacking Materials Correctly
Bonded layers of dissimilar materials don't expand independently. They argue. A bi-metal strip is the textbook case, and the same physics shows up in clad pipe, glass-to-metal seals, ceramic coatings on superalloys, and silicon dies on copper lead frames.
A bi-metal strip consists of two metals with different CTE values bonded together. When heated, the strip bends because one side expands more than the other. The constrained expansion sets up an internal stress field, and the strip resolves it by curving. This principle drives thermostats, temperature indicators, and compensation mechanisms.
Curvature: κ = 6(α₁ - α₂)(T - T₀) / h
where h = total strip thickness, assuming equal layer thickness
Deflection (cantilever): δ = κL²/2
where L = free length of strip
| Common Bi-Metal Combinations | High-CTE Layer (α ×10⁻⁶/°C) | Low-CTE Layer (α ×10⁻⁶/°C) | Application |
|---|---|---|---|
| Brass/Steel | 19 | 12 | Thermostats |
| Brass/Invar | 19 | 1.3 | High sensitivity |
| Copper/Steel | 16.5 | 12 | Circuit breakers |
For axially constrained stacks, you can also build a CTE-matched assembly. Clock pendulums use a gridiron design (alternating brass and steel rods) so vertical expansion in brass cancels contraction effects from steel, holding pendulum length constant against ambient swings. Glass-to-metal seals in vacuum tubes rely on Kovar (α ≈ 5.5) deliberately matched to borosilicate glass, because mismatch above 1 ppm/°C cracks the seal on cooldown.
Stacking rule of thumb: If two layers must stay bonded across a 100°C swing, target Δα below 5 ×10⁻⁶/°C. Ceramics on metals often need a graded interlayer because Δα is too large to absorb directly.
Steady-State vs. Transient: When Each Applies
Plug ΔT into ΔL = αL₀ΔT and you've assumed the part is uniformly at the new temperature. That's steady-state thinking. Real parts have a thermal time constant τ ~ L²/α_diff, where α_diff is thermal diffusivity (k/(ρc_p), units m²/s). Aluminum has α_diff ≈ 9.7×10⁻⁵ m²/s, carbon steel ≈ 1.2×10⁻⁵, fused silica ≈ 8×10⁻⁷. A 50 mm aluminum block reaches 95% of its final temperature in roughly 30 seconds in still water; a 50 mm fused silica disk takes about an hour.
The transient regime matters when the thermal load changes faster than the part can equilibrate. A turbine blade going through a startup ramp sees its outer skin heat first while the core lags, and the skin is forced to expand against a colder, stiffer interior. That mismatch drives thermal-shock failure.
Use steady-state ΔL = αL₀ΔT when
- Heating/cooling slow vs. τ (e.g., daily ambient swings on rail)
- Part is small or highly conductive (Biot < 0.1)
- Final equilibrium temperature is what matters
- Stress isn't the design driver, just dimensional change
Switch to transient analysis when
- Step changes in surface temperature (quench, startup)
- Internal gradients drive failure (thermal shock)
- Cycle period is shorter than τ (engine warm-up)
- Massive walls (concrete, masonry) under daily load swings
A practical check: compute Biot number Bi = hL/k. If Bi < 0.1, the part is roughly isothermal at any instant and lumped-capacitance analysis works. Above 0.1, internal gradients matter and you can't treat ΔT as uniform.
Material Properties That Drive the Answer (k, α, c, ε)
The linear coefficient of thermal expansion (α or CTE) quantifies how a material's length changes per degree of temperature change. It's defined as α = (1/L₀) × (dL/dT) with units of 1/°C or 1/K, commonly expressed as ×10⁻⁶/°C (ppm/°C or μm/m/°C). But α isn't the only property you'll need on the bench. Thermal conductivity k decides how fast heat penetrates. Specific heat capacity c sets the energy needed for a given ΔT. Surface emissivity ε governs radiative gain or loss.
Low CTE Materials
- Fused silica: 0.55
- Invar 36: 1.3
- Tungsten: 4.5
- Silicon: 2.6
Used for precision instruments, optics
Medium CTE Materials
- Steel: 10 to 13
- Cast iron: 10 to 12
- Titanium: 8.6
- Copper: 16.5
Structural and general engineering
High CTE Materials
- Aluminum: 23.6
- Zinc: 30.2
- Lead: 28.9
- Polymers: 50 to 200
Require expansion allowance
Material families show characteristic ranges: metals typically fall between 10 and 25 ×10⁻⁶/°C, ceramics and glasses between 0.5 and 10, polymers between 50 and 200. Within each family, atomic bonding strength and crystal structure decide the specific number. Strong covalent bonds (silicon carbide, fused silica) resist expansion; weak van der Waals interactions in polymers permit large dimensional changes per degree.
| Material | α (×10⁻⁶/°C) | Expansion per meter per 100°C |
|---|---|---|
| Aluminum alloys | 21 to 24 | 2.1 to 2.4 mm |
| Brass | 18 to 21 | 1.8 to 2.1 mm |
| Bronze | 17 to 18 | 1.7 to 1.8 mm |
| Carbon steel | 10.8 to 12.5 | 1.1 to 1.25 mm |
| Stainless 304 (austenitic) | 16 to 18 | 1.6 to 1.8 mm |
| Copper (C11000) | 16.5 | 1.65 mm |
| Titanium alloys | 8.0 to 9.0 | 0.8 to 0.9 mm |
| PTFE (Teflon) | 100 to 150 | 10 to 15 mm |
| HDPE | 100 to 200 | 10 to 20 mm |
| Borosilicate (Pyrex) | 3.2 to 3.5 | 0.32 to 0.35 mm |
| Fused silica | 0.5 to 0.6 | 0.05 to 0.06 mm |
For isotropic materials, area and volume coefficients link directly to α: β = 2α and γ = 3α. The factors 2 and 3 arise because area involves two linear dimensions and volume involves three. The approximation holds while αΔT < 0.01, meaning the dimensional change stays under 1%. For aluminum at α = 23.6×10⁻⁶/°C this corresponds to ΔT < 420°C, valid for almost every engineering case.
Hole expansion: When sizing a hole for thermal fit, use the linear coefficient on the diameter. A steel plate at 20°C with a 10.000 mm hole heated to 200°C grows by ΔD = α × D₀ × ΔT = 11.7×10⁻⁶ × 10 × 180 = 0.021 mm. Holes expand the same way solid material does, not the opposite.
CTE itself varies with temperature. Aluminum sits at 23.1 at 20°C, 25.4 at 200°C, 28.5 at 500°C. Invar's ultra-low CTE only holds near room temperature; above 200°C, it behaves like ordinary nickel-iron with α near 5, and by 500°C it's up around 10.5. Use mean coefficients over the actual operating range, or integrate ∫α(T)dT for precision work. NIST and ASM publish polynomial fits for the common engineering metals.
Note on anisotropy: Some materials have different CTE values along different axes. Graphite can expand 10× more perpendicular to the basal plane than parallel. Wood, fiber composites, and single crystals all exhibit directional expansion that requires axis-specific coefficients.
Real-World Conditions: Cold Bridges, Edge Effects, Surface Resistance
Lab values assume free, uniform expansion. Field hardware doesn't cooperate. Anchored ends, bolted flanges, mismatched supports, and uneven solar loading all distort the simple ΔL = αL₀ΔT picture, and you have to back the corrections out by inspection.
Differential expansion: clearance grows when
- αouter > αinner on heating
- αouter < αinner on cooling
- Aluminum housing, steel shaft: clearance grows when heated
Differential expansion: clearance shrinks when
- αouter < αinner on heating
- αouter > αinner on cooling
- Steel housing, aluminum shaft: clearance shrinks when heated
Cold bridges are the structural analog of thermal shorts: a steel beam passing through an insulated wall conducts heat (and temperature) faster than its surroundings, so it sits at a different ΔT than the bulk and expands or contracts on its own schedule. Edge effects matter at flange faces and bolted joints where constraint is partial. Surface resistance (the air-film effect) decouples the surface temperature from the air temperature, so a sun-exposed steel rail can run 20°C above shade ambient even with breeze.
Shrink Fit Design
Shrink fits exploit differential expansion. Heat the outer part to expand the bore, insert the inner part at room temperature, then cool. Interference on cooldown creates a tight assembly without mechanical fasteners. Typical interference: 0.001 to 0.002 mm per mm of diameter. Required ΔT = (interference + clearance)/(α × diameter).
Limitations and Assumptions
Constant CTE: Calculations assume α stays fixed across the temperature range. For swings exceeding 100°C, use mean coefficients or integrated values from material property databases.
Isotropic material: Formulas assume uniform expansion in all directions. Composites, wood, single crystals, and rolled metals may show direction-specific α requiring axis-by-axis analysis.
Free expansion only: The calculator computes geometric change for unconstrained parts. Constrained expansion develops thermal stress instead of dimensional change; stress analysis is a separate step using σ = EαΔT for fully restrained members.
Uniform temperature: Calculations assume instant, uniform body temperature. Real heating and cooling produce thermal gradients, transient stresses, and non-uniform expansion that need transient FEA for precision.
Educational Use Only: The calculator demonstrates expansion principles with reference values from standard sources. Professional thermal design needs certified material data, stress analysis for constrained systems, fatigue assessment under cycling, and code compliance (ASME, AISC, AWS). Consult qualified engineers for structural applications.
Worked Example: 30 m Steel Rail Through a 50°C Annual Swing
Problem: A 30 m carbon-steel rail section is welded into track at 15°C. The rail material has α = 12×10⁻⁶/°C. Winter low at the site is -10°C and summer high is +40°C. What free expansion does the rail try to undergo, and what does that imply for joint or anchorage design?
Step 1: Pin down the temperature range
ΔT_summer = 40 - 15 = +25°C (rail wants to grow)
ΔT_winter = -10 - 15 = -25°C (rail wants to shrink)
Total swing about installation: ΔT_total = 50°C peak-to-peak
Step 2: Compute peak free expansion at +40°C
ΔL_summer = α × L₀ × ΔT_summer
ΔL_summer = 12×10⁻⁶ × 30 × 25 = 0.0090 m = 9 mm
Step 3: Compute peak free contraction at -10°C
ΔL_winter = 12×10⁻⁶ × 30 × (-25) = -0.0090 m = -9 mm
Step 4: Total free movement across the year
ΔL_total = α × L₀ × ΔT_total = 12×10⁻⁶ × 30 × 50 = 0.018 m = 18 mm
Step 5: Translate to design intent
If the rail is jointed, install with a 9 mm gap centered on the swing range so summer never closes the gap and winter doesn't over-open it. If continuously welded (CWR), the rail isn't free to grow; it builds compressive stress σ = EαΔT = 200 GPa × 12×10⁻⁶ × 25 = 60 MPa in summer and equal tension in winter, and the anchorage and ballast must resist that load to prevent sun-kink buckling.
Bridge bearings tell the same story. A 30 m simply-supported steel girder over the same temperature range moves 18 mm. Sliding bearings or elastomeric pads at one end take that displacement so the abutments aren't loaded by thermal force. Skip the bearing and the next thing to fail is either the bolts or the abutment concrete.
References
Reliable thermal expansion data needs authoritative sources. Verify CTE values against published standards and material-specific data sheets before any structural calculation. Measurement uncertainty for metals typically runs ±2 to 5%; alloy composition and heat-treatment state shift α by similar amounts.
- ASM Handbook, Volume 1. Properties and Selection: Irons, Steels, and High-Performance Alloys. ASM International. Reference for ferrous metal properties including temperature-dependent α.
- ASM Handbook, Volume 2. Properties and Selection: Nonferrous Alloys and Special-Purpose Materials. ASM International. Aluminum, copper, titanium, and specialty alloys.
- NIST Standard Reference Database (SRD 69, SRD 100). Thermophysical Properties of Matter, including JANAF Thermochemical Tables. nist.gov/srd
- ASTM E228-22. Standard Test Method for Linear Thermal Expansion of Solid Materials by Push-Rod Dilatometry (metals, ceramics).
- ASTM E831. Thermomechanical Analysis (TMA) for polymers and films.
- ASTM D696-16. Coefficient of Linear Thermal Expansion of Plastics by Vitreous Silica Dilatometer.
- ISO 11359-2. Plastics, Thermomechanical Analysis, Determination of Coefficient of Linear Thermal Expansion.
- ASHRAE (2021). Handbook of Fundamentals. American Society of Heating, Refrigerating and Air-Conditioning Engineers. Chapter 39 ("Building Envelopes") and the materials chapter list α values for masonry, concrete, glass, and insulation that drive joint-spacing calculations in building enclosures.
- Incropera, F. P. & DeWitt, D. P. Fundamentals of Heat and Mass Transfer (8th ed., 2017). Wiley. The standard textbook treatment of conduction, convection, and radiation that drives thermal loads.
- Holman, J. P. Heat Transfer (10th ed., 2010). McGraw-Hill. Engineering reference with worked examples on transient conduction and Biot-Fourier analysis.
- Callister, W. D. & Rethwisch, D. G. (2020). Materials Science and Engineering: An Introduction (10th ed.). Wiley. Bonding-based explanation for the systematic CTE differences between metals, ceramics, and polymers.
- MatWeb Material Property Data. matweb.com. Database of over 130,000 materials with CTE entries.
Reference values current as of February 2026. Material properties shift with alloy composition, heat treatment, and processing route. Verify against material certifications for critical applications.
Troubleshooting Thermal Expansion Problems and Material Selection
Real questions from engineers and designers stuck on differential expansion failures, shrink fit calculations, CTE data discrepancies, and why their assemblies loosen up at operating temperature.
What is the coefficient of thermal expansion?
The coefficient of thermal expansion (CTE) is the fractional change in length per degree of temperature change: α = (1/L) · (dL/dT). Units are typically 1/K or 1/°C, often written as ppm/°C (parts per million per degree). A material with α = 12 ppm/°C grows by 12 micrometers per meter per degree. For practical use, ΔL = α · L · ΔT for linear expansion. Volume expansion uses β ≈ 3α for isotropic solids. A 20-foot (6.1 m) aluminum gutter (α = 23 × 10⁻⁶ /°C) heated from −10°C to 40°C grows by 6.1 · 23 × 10⁻⁶ · 50 = 7 mm. That's enough to buckle a rigidly mounted run, which is why expansion joints exist. Common values: structural steel 12, concrete 10, copper 17, aluminum 23, glass 9 (regular soda-lime), borosilicate glass 3.3, Invar 1.2, all in ppm/°C. Polymers run much higher (PVC 70, polyethylene 200). The reason borosilicate (Pyrex) lab glass survives thermal shock is the low α: less stress per degree. CTE matters anywhere two materials get bonded or constrained: PCB-to-component joints, bridge expansion gaps, train-track welds, spacecraft optics. Mismatched CTEs build stress on every thermal cycle, eventually fatiguing the joint. Bimetallic strips turn this into a feature: the differential bend with temperature drives mechanical thermostats. The coefficient itself isn't perfectly constant. It varies with temperature for most materials and changes discontinuously at phase transitions.
Our piping engineer says stainless steel expands more than carbon steel. That seems wrong since they're both steel. Is he right?
Your engineer is correct. Austenitic stainless steels (304, 316) have CTE around 17×10⁻⁶/°C, while carbon steel is about 11.7×10⁻⁶/°C—stainless expands roughly 45% more. This is because austenitic stainless has a different crystal structure (face-centered cubic) than carbon steel (body-centered cubic). Mixing stainless and carbon steel piping requires careful expansion analysis. Ferritic and martensitic stainless steels have CTE closer to carbon steel.
I installed a 20-foot aluminum gutter in winter at 35°F. Now it's buckled in summer. How much does aluminum expand over that temperature range?
Aluminum has a CTE of about 23.6×10⁻⁶/°C. A 20-foot (6.1 m) gutter going from 35°F (2°C) to 95°F (35°C) expands by ΔL = 23.6×10⁻⁶ × 6.1 × 33 = 0.0047 m = 4.7 mm, roughly 3/16 inch. Aluminum gutters over 10 feet need expansion joints or sliding connections. Without them, the expansion force buckles the material. Each 10 feet of aluminum gutter needs roughly 2.4 mm (3/32") expansion allowance for a 30°C swing.
I'm pressing a steel shaft into an aluminum housing. At room temperature there's 0.05 mm interference. Will it loosen up at operating temperature of 100°C?
Yes, it will loosen because aluminum expands faster than steel. For a 50 mm diameter: Steel shaft growth = 11.7×10⁻⁶ × 50 × 80 = 0.047 mm. Aluminum housing bore growth = 23.6×10⁻⁶ × 50 × 80 = 0.094 mm. Net clearance change = 0.094 - 0.047 = 0.047 mm. Your initial 0.05 mm interference reduces to only 0.003 mm—essentially a loose fit. For aluminum housings with steel shafts operating hot, increase room-temperature interference or consider a mechanical retention method.
My textbook gives two different CTE values for aluminum: 23.1 and 23.6. Which one is correct?
Both can be correct—CTE varies with alloy and temperature range. Pure aluminum is about 23.1×10⁻⁶/°C at 20°C; 6061-T6 is 23.6×10⁻⁶/°C averaged from 20–100°C. CTE also increases at higher temperatures: aluminum at 500°C has CTE around 28×10⁻⁶/°C. Always check the specific alloy and temperature range in your reference. For critical applications, request actual test data from the material supplier rather than handbook values.
We're designing a bridge expansion joint for a 100-meter steel span. Our junior engineer calculated 120 mm of movement. That seems too much—did he make an error?
His calculation is in the right ballpark. For carbon steel (α = 11.7×10⁻⁶/°C) with a temperature swing from -20°C to +40°C (ΔT = 60°C): ΔL = 11.7×10⁻⁶ × 100 × 60 = 0.070 m = 70 mm. If he used a wider range like -30°C to +50°C (ΔT = 80°C), he'd get 94 mm. Factor in creep, concrete shrinkage, and safety margin, and 120 mm total joint capacity is reasonable. The 120 mm might be the specified joint capacity rather than the calculated expansion alone.
I found a brass/steel bi-metal strip in old equipment. How do these work and why would someone use them instead of electronics?
Bi-metal strips exploit differential expansion: brass (CTE ≈ 19×10⁻⁶/°C) and steel (CTE ≈ 12×10⁻⁶/°C) are bonded together. When heated, brass expands more, causing the strip to bend toward the steel side. This mechanical motion triggers switches in thermostats, circuit breakers, and temperature indicators. They're still used where electronics aren't practical: high-temperature environments, explosive atmospheres, fail-safe systems, or where simplicity and reliability matter more than precision. They never need batteries or calibration.
Our railroad maintenance crew says they need to leave 12 mm gaps between 25-meter rails, but I calculated only 11.7 mm. Are they over-engineering this?
Your calculation is probably based on temperature only: ΔL = 11.7×10⁻⁶ × 25 × 40 = 11.7 mm for a 40°C swing. The crew is right to add margin. Real-world factors include: installation temperature uncertainty, residual stresses from rolling, track curvature effects, and that rail buckling is catastrophic while slightly large gaps just cause noise. Also, standard rail joiner fishplates have specific gap specifications. Modern continuously welded rail eliminates gaps entirely by anchoring rails and allowing them to develop internal stress instead of expanding freely.
I need my laser cavity to stay dimensionally stable within 0.001 mm over a 5°C temperature variation. What material should I use?
For that level of stability, you need ultra-low expansion material. Options: Invar 36 (CTE ≈ 1.3×10⁻⁶/°C at room temperature), Super Invar (CTE ≈ 0.6×10⁻⁶/°C), Zerodur glass-ceramic (CTE ≈ 0.02×10⁻⁶/°C), or fused silica (CTE ≈ 0.55×10⁻⁶/°C). For a 100 mm cavity over 5°C: Zerodur gives 0.00001 mm change; Invar gives 0.00065 mm. Your 0.001 mm tolerance requires Zerodur or active temperature control with Invar. Note: Invar's low CTE only holds near room temperature—it's not suitable for high-temperature applications.
Our HDPE pipe supplier says to allow 15 mm/m for thermal expansion. That's 10x more than copper. Is that right?
Yes, polymers have much higher CTE than metals. HDPE ranges from 100–200×10⁻⁶/°C compared to copper's 16.5×10⁻⁶/°C. For a 1-meter HDPE pipe with ΔT = 100°C: ΔL = 150×10⁻⁶ × 1 × 100 = 0.015 m = 15 mm. This is why plastic piping systems require many more expansion loops, guides, and anchors than metal systems. Ignoring polymer expansion is a common cause of buckled pipes, broken fittings, and wall penetration damage. Always follow manufacturer expansion guidelines for plastic piping.
My physics teacher says to use β = 2α for area expansion, but when I calculate it exactly for a square plate, I get a slightly different answer. Who's wrong?
You're both right, depending on precision needed. The exact formula for a square: A₂ = L²(1 + αΔT)² = A₁(1 + 2αΔT + α²ΔT²). The β = 2α approximation ignores the (αΔT)² term. For steel (α = 12×10⁻⁶/°C) at ΔT = 100°C, αΔT = 0.0012. The squared term is (0.0012)² = 0.0000014, which is negligible. But for polymers (α = 150×10⁻⁶/°C), αΔT = 0.015, and the squared term is 0.000225—a 0.02% error. For engineering purposes, β ≈ 2α is almost always sufficient; the exact formula only matters for extreme precision or very high expansion materials.