Thermal expansion: one coefficient, and the 2α and 3α that follow from it

Q: Should I use the original length or the final length in the formula?

The original, L0, as written. The difference between using L0 and using the expanded L is itself of order αΔT — a few hundred parts per million of a quantity that is already a few hundred parts per million — so it is a correction of order 10−7 on the length and nobody can measure it. It becomes real only in the same regime where the exact cube separates from 3αΔT, which is strains of a per cent or more, and at that point the bigger problem is that α is not constant over the range anyway.

α for length, 2α for area, 3α for volume — from a single measured coefficient. This page gives all three, the exact cube beside the 3αΔT approximation so you can see when the approximation stops being safe, and why a hole in a plate gets bigger.

Last updated 11 Nov 2026

Expansion in length, area and volume

One α, three coefficients, and the exact cube beside the approximation

Original length (or the side of the square or cube)In metres. The area and volume rows below treat this as the side of a square and of a cube, which is the cleanest way to show how α, 2α and 3α relate — but the coefficients themselves do not care about shape at all. A sphere, a girder and a hole in a plate all expand by the same fractional amount per kelvin.

Temperature changeA change, in degrees, and the same number in Celsius or kelvin. Negative means cooling and gives a negative expansion, which is a contraction. A railway rail in Britain might see 40 K between a winter night and a sunlit summer afternoon; in central India a steel bridge can see 50 or more, and the surface temperature of dark metal in sun runs well above the air temperature.

Coefficient of linear expansion αSeven values, all read from one published table on 11 November 2026, and all at about 25 °C. Notice what is missing: copper, brass, steel and concrete. The same table gives them as RANGES rather than single numbers — copper 16 to 16.7, brass 18 to 19, steel 10.8 to 12.5, concrete 13 to 14 — and it says plainly that the spread may come from the material varying or from sources disagreeing. Printing a midpoint would hide that, so they are not offered: type your question’s value in the last option instead. The pair worth comparing is ordinary glass at 9.0 against Pyrex at 4.0, which is the whole reason laboratory glassware is borosilicate: half the expansion means half the thermal stress when you pour boiling water into it.

Coefficient of linear expansion, typed inLocked unless the last option is chosen above. In units of 10⁻⁶ per kelvin, which is how every table prints it: type 23 for aluminium, not 0.000 023. Per °C and per K are the same number. A value in 10⁻⁶ per °F would need multiplying by 1.8.

12.000mmExample

A 25 m iron rail warmed by 40 K, on α = 12.0 x 10^-6 per K

Why area goes as 2α and volume as 3α, and where that stops being exact

ΔL = L₀ α ΔT · β = 2α · γ = 3α · exactly: V/V₀ = (1 + αΔT)³ = 1 + 3αΔT + 3(αΔT)² + (αΔT)³

α: the coefficient of linear expansion: the fractional change in length per kelvin. Typical metals are tens of parts per million per kelvin, so tables print it as a number times 10⁻⁶. It is a property of the material, mildly dependent on temperature, and the only measured quantity in this whole page
β = 2α: the area coefficient. An area is two lengths multiplied, each growing by the factor (1 + αΔT), so the area grows by that factor squared — and (1+x)² = 1 + 2x + x², which is 1 + 2x whenever x is small. For αΔT of a few hundred parts per million the x² term is a few parts in 10⁸
γ = 3α: the volume coefficient, by the same argument cubed. The three coefficients are not independent constants to be looked up separately: for an isotropic solid there is ONE measured number and the other two follow. Using α where γ belongs understates a volume expansion by a factor of three, which is the error this page exists to stop
a hole: a hole in a plate expands exactly as the material does, which is the result everyone finds counter-intuitive. Draw the hole’s outline on the solid plate: every point on that outline moves outward with the metal around it, and nothing about removing the inside changes where those points go. So a ring warmed to free a stuck bolt works, and a gap between two rails closes as both rails grow
ΔT: a temperature DIFFERENCE, so Celsius and kelvin give the same number and α per °C equals α per K. A coefficient quoted per degree Fahrenheit is 1.8 times smaller for the same material, which is the only unit trap on this page

Worked example

A 25 m iron rail warmed by 40 K, on α = 12.0 x 10^-6 per K
THE STRAIN FIRST, because it is the quantity that does not depend on the size of the object. αΔT = 12.0 × 10⁻⁶ × 40 = 4.8 × 10⁻⁴, which is 480 parts per million or 0.048 per cent. Every iron object in that temperature change grows by that fraction, from a rivet to a bridge
THE LENGTH CHANGE. ΔL = 25 × 4.8 × 10⁻⁴ = 0.012 m = 12.000 mm. Worth holding on to as a feel for the scale: a 25 m rail moves about a centimetre over a British summer's temperature swing, which is why jointed track was laid with gaps and why continuous welded rail has to be stressed when it is installed
AREA, ON THE SAME α. A square of side 25 m has area 625 m². The exact new area is 625 × (1.000 48)² = 625.600 144 m², a change of 0.600 144 m². The 2αΔT approximation gives 625 × 2 × 4.8 × 10⁻⁴ = 0.600 000, low by 0.024 per cent of the expansion. For a school answer they are the same number
VOLUME, WHERE THE FACTOR OF THREE LIVES. A cube of side 25 m has volume 15 625 m³. Exactly: 15 625 × (1.000 48)³ gives a change of 22.510 8 m³, against 3αΔT's 22.500 0 — low by 0.048 per cent. Compare that with the mistake of using α instead of 3α, which would give 7.5 m³ and be low by 67 per cent. The approximation is harmless; the wrong coefficient is not
AND THE SENSITIVITY. To grow this rail by exactly 1 mm takes ΔT = 0.001 / (25 × 12 × 10⁻⁶) = 3.333 K. That number is why expansion matters in engineering and almost never in a kitchen: the effect is small per degree and the objects are long, so it is length, not temperature, that makes it visible

A 25 m rail warmed by 40 K: the same sum with each coefficient

The third column is numerically the same as the second here, which is a coincidence of the chosen numbers worth noticing: 25 m × 40 K = 1000 m·K, so the expansion in millimetres is exactly the coefficient in parts per million. That makes the table easy to read and the ranking obvious — lead moves nineteen times as far as Invar for the same warming. The fifth column is the one that matters on a building site: a 25 m iron member needs only 3.3 K to grow a millimetre, which is less than the difference between shade and sunlight on the same afternoon. Copper, brass, steel and concrete are absent because the source table gives them as ranges (16–16.7, 18–19, 10.8–12.5 and 13–14) and states that the spread may be real variation or source disagreement; a midpoint invented here would look more certain than the data is.
Material	α (10⁻⁶/K)	Expansion of 25 m over 40 K	Strain (ppm)	ΔT for exactly 1 mm	Volume coefficient 3α (10⁻⁶/K)
Aluminium	23	23.00 mm	920	1.74 K	69
Lead	29	29.00 mm	1160	1.38 K	87
Pure iron	12.0	12.00 mm	480	3.33 K	36
Forged iron	11.3	11.30 mm	452	3.54 K	33.9
Glass, plate	9.0	9.00 mm	360	4.44 K	27
Pyrex	4.0	4.00 mm	160	10.00 K	12
Invar	1.5	1.50 mm	60	26.67 K	4.5

One measured coefficient, two that follow from it, and a hole that gets bigger

There is only one measured number in thermal expansion, and the other two are arithmetic. A solid’s coefficient of linear expansion α says what fraction of its length it gains per kelvin. An area is two lengths multiplied, so it grows by (1 + αΔT) squared; a volume is three, so it grows by the cube. Expand those brackets and, for the small values αΔT actually takes, you get β = 2α and γ = 3α. They are not three independent constants to be looked up: for an isotropic solid they are one constant and two multiplications.

The mistake that costs marks is using α where γ belongs. It understates a volume expansion by a factor of three, and the wrongness is invisible in the working because the arithmetic is correct. The reverse slip, using 3α for a length, overstates by three. This page prints all three coefficients at once for exactly that reason, and prints the exact cube alongside the approximation so you can see which of the two kinds of error actually matters. For a 25 m iron rail warmed 40 K, the approximation is low by 0.048 per cent of the expansion and using the wrong coefficient would be low by 67 per cent.

A hole in a plate expands exactly as the material does. This is the result that feels wrong and is worth settling once. Draw the outline of the hole on the solid plate before anything is cut out. Warm the plate: every point on that line moves outward along with the metal surrounding it. Now cut the inside away. Nothing you removed was telling the outline where to go, so the hole is the same size it would have been. That is why warming a ring frees a stuck bolt, why a steel tyre is shrunk onto a wheel by heating it first, and why a gap left between two rails closes from both sides.

Where the numbers come from, and what is deliberately not here. The seven coefficients in the dropdown were read from one published table, at about 25 °C. Copper, brass, steel and concrete are missing from it on purpose: the same table gives them as ranges — 16 to 16.7, 18 to 19, 10.8 to 12.5, 13 to 14 — and says the spread may be the material varying or the sources disagreeing. A single midpoint printed here would look more certain than the data is, so they go in the typed field instead, with your question’s own value. The pair worth comparing is in the list though: ordinary plate glass at 9.0 against Pyrex at 4.0, which is the entire reason laboratory glassware is borosilicate.

And what this page deliberately does not calculate: thermal stress. If a material is free to expand, it expands and nothing happens. If it is held, it cannot, and the strain it was going to have becomes a stress instead — which is what cracks concrete, buckles rail and breaks glassware. That calculation needs the material’s Young modulus and belongs to engineering rather than to school thermal physics, and this family of sites already covers it on the mechanical side. What belongs here is the strain itself, which is printed above in parts per million, and the understanding that a restrained 0.048 per cent is a great deal of stress in steel.

Frequently asked questions

Why is the coefficient of volume expansion three times the linear one?

Because a volume is three lengths multiplied together and each of them grows by the same factor. Exactly, V/V₀ = (1 + αΔT)³ = 1 + 3αΔT + 3(αΔT)² + (αΔT)³. Since αΔT is typically a few hundred parts per million, the squared term is a few parts in 10⁸ and the cubed term is negligible beyond writing down, so γ = 3α to far better precision than any coefficient is known. The same argument gives β = 2α for area. It holds for isotropic materials; a crystal or a composite can expand differently along different axes and then needs a coefficient per direction.

Does a hole in a metal plate get bigger or smaller when the plate is heated?

Bigger, by exactly the fraction the material expands. The cleanest argument is to imagine the hole’s outline drawn on the uncut plate: when the plate warms, every point of that outline moves outward with the metal around it, and cutting the inside away afterwards changes nothing about where those points went. So the hole’s diameter grows by αΔT, its area by 2αΔT, and a spherical cavity’s volume by 3αΔT. It is the principle behind warming a ring to free a bolt and behind shrink fitting.

Why is laboratory glassware made of Pyrex?

Because its coefficient of expansion is less than half ordinary glass’s — 4.0 against 9.0 × 10⁻⁶ per kelvin in the table this page reads — and thermal shock failure comes from different parts of an object wanting to be different sizes at once. Pour boiling water into a cold vessel and the inside surface tries to expand while the outside has not warmed yet; the inside is restrained, so it goes into compression and the outside into tension, and glass fails in tension. Halving the coefficient roughly halves that stress for the same temperature shock. It is not about melting point or strength.

Is α really constant?

No, it varies with temperature, and every table that prints a single figure is quoting a value for a stated range — about 25 °C for the figures used here. For metals near room temperature it changes slowly enough that treating it as constant over a few tens of kelvin is well inside the precision of a school answer. Over hundreds of kelvin it is not, and for a few materials it does something much more dramatic: water between 0 and 4 °C has a NEGATIVE volume coefficient, so it is densest at about 4 °C and a pond’s coldest water sits at the top rather than the bottom. Invar’s whole purpose is a near-cancellation that works only over a limited range.

Should I use the original length or the final length in the formula?

The original, L₀, as written. The difference between using L₀ and using the expanded L is itself of order αΔT — a few hundred parts per million of a quantity that is already a few hundred parts per million — so it is a correction of order 10⁻⁷ on the length and nobody can measure it. It becomes real only in the same regime where the exact cube separates from 3αΔT, which is strains of a per cent or more, and at that point the bigger problem is that α is not constant over the range anyway.

Related calculators

References

Linear Thermal Expansion Coefficients of Materials, The Engineering ToolBox (2003) (read 11 November 2026). The source of all seven coefficients in this page’s dropdown, stated by the page as values at about 25 °C (77 °F): aluminium 23, lead 29, pure iron 12.0, forged iron 11.3, ordinary plate glass 9.0, Pyrex/borosilicate 4.0 and Invar 1.5, all in 10⁻⁶ per K. It also gives copper as 16–16.7, brass 18–19, steel 10.8–12.5, cast iron 10.4–11 and concrete 13–14, and states that a spread may reflect variation in the material or disagreement between sources — which is why this page offers only the single-valued entries and asks for anything else. The page names no primary source for its own figures and cites only itself; that limitation is stated here rather than hidden, and it is the reason the typed field exists.
COULD NOT BE READ, and nothing was substituted for any of it. The NCERT textbook pages on ncert.nic.in, and four CIET modules in the same series as the ones cited here — Keph_201101 (temperature and its measurement), Keph_201102 (thermal expansion), Keph_201103 (specific heat capacity and calorimetry) and Keph_201105 (conduction and convection) — were all refused by the server’s robots policy on 11 November 2026. That is why this batch quotes no NCERT table number, and why the coefficients of expansion and the thermal conductivities are attributed to the reference actually read rather than to the textbook table you are more likely to be holding. If your table disagrees, use your table: these pages take the constant from you for exactly that reason.
The International System of Units (SI), 9th edition, BIPM, published May 2019; the version read for this batch is 4.01, revised June 2026 (read 11 November 2026). The source of the kelvin’s CURRENT definition: the kelvin is defined by fixing the Boltzmann constant at exactly k = 1.380 649 × 10⁻²³ kg m² s⁻² K⁻¹, which the brochure states as an exact relation. The degree Celsius is defined equal in magnitude to the kelvin, with T₀ = 273.15 K and t/°C = T/K − 273.15. Most school textbooks still define the kelvin as 1/273.16 of the triple point of water. The brochure records that as the PREVIOUS definition, and says the triple-point temperature must now be determined experimentally — it stood at 273.16 K with a relative standard uncertainty of 3.7 × 10⁻⁷ at the redefinition, and is a measured quantity from then on.
NCERT / CIET, the Physics-02 module series for Unit 7, “Properties of Bulk Matter” (read 11 November 2026). Cited only to place this page in the CBSE class XI syllabus, not for a number. The modules that could be read identify thermal expansion, calorimetry, change of state, conduction, convection and radiation as chapter 11, “Thermal Properties of Matter”, of Physics Part II. The module whose file is keph_201106_eContent2019.pdf — its own heading reads Keph_201105, so the file name and the heading disagree by one — is Module 17, “Radiation (Black body)”, and lists “coefficient of thermal conductivity” as a Module 16 topic while giving no numerical value for it. That module also gives the Stefan constant as 5.6704 × 10⁻⁸ W m⁻² K⁻⁴, Wien’s constant as 2.88 × 10⁻³ m K, and Newton’s law of cooling as −dQ/dt = k(T₂ − T₁), valid only for small temperature differences.