Skip to content
Calcapedia
Menu
Science

Thermal Expansion Calculator

See how much a beam, pipe or tank of liquid grows or shrinks when it warms or cools, with typical values for common materials.

Level: Grades 10–11

Printable worksheet

Thermal Expansion Calculator

Know the answer you need? Pick an input to work backward.

Inputs
Expansion type

Typical values near room temperature; grades and alloys vary.

Final temperature minus starting temperature. Negative for cooling (contraction).

Result

Change in length (ΔL)

14.4 mm

The length changes by 14.4 mm.

Change in meters
0.0144 m
Final length
30.0144 m
Change as a percentage of L₀
0.048%
Coefficient used (α)
12 × 10⁻⁶ /K

See the step-by-step solution ↓

Step-by-step solution

  1. Look up the typical expansion rate for carbon steel: how much of its size it grows per degree

    α=12×10−6=0.000012 /K

  2. Multiply the expansion rate by the starting length and the temperature change to get the growth

    0.000012×30×40=0.0144 m

  3. Change the result into the unit you picked, mm (1 m = 1,000 mm)

    0.0144 m×1,000=14.4 mm

  4. Add the change to the starting length to get the new length (L = L₀ + ΔL)

    30+0.0144=30.0144 m

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out how much something grows when it warms up, or shrinks when it cools. Almost every material does this. The effect is tiny per degree, often a few millionths of the size. But across a long bridge, a run of pipe or a tank of fuel, it adds up to millimeters, centimeters or liters. That's why bridges have expansion joints. It's also why fuel tanks are never filled to the brim.

The calculator works out the change for a length (linear expansion) or a volume (volumetric expansion). Each material has its own expansion rate, called its coefficient: how much it grows per degree. Choose a material to use a typical rate, or enter your own from a data sheet. Then give the starting size and the temperature change.

Where it's used

Why this math matters outside the classroom:

  • Bridge joints. Bridge builders leave expansion joints because a 100 m steel span grows about 100 × 12 × 10⁻⁶ × 40 = 4.8 cm over a 40 °C swing.
  • Railway track. Rail engineers stress or space track so rails don't buckle on hot summer days.
  • Stuck jar lids. Running a stuck metal lid under hot water loosens it because the metal expands more than the glass.
  • Thermometers. Liquid thermometers work because the liquid expands by a predictable amount with each degree.

How to use it

  1. Choose linear expansion for a length, such as a rail, beam or pipe. Choose volumetric expansion for a volume of liquid or solid.
  2. Pick a material, or choose to enter your own expansion rate and its unit.
  3. Enter the starting length or volume and the temperature change. Use a negative change for cooling.
  4. Choose the unit for the change. The final size and the percentage change are listed underneath.

What the inputs mean

Material
The listed rates are typical values near room temperature. Steel alloys, concrete mixes, glasses and plastics vary by grade, sometimes by 20% or more. So use a manufacturer's figure for design work.
Linear expansion coefficient (α)
Tables usually list α in millionths (10⁻⁶) per kelvin. That's also written µm/(m·K) or ppm/°C. A kelvin step is the same size as a Celsius degree. A rate given per °F is 5/9 of the same rate per °C.
Temperature change (ΔT)
Enter how much the temperature changes, not the final temperature. A change of 1 °C equals a change of 1 K; a change of 1 °F is 5/9 K.

Formula and method

ΔL = α × L₀ × ΔT
ΔV = β × V₀ × ΔT
L = L₀ + ΔL;  V = V₀ + ΔV
β ≈ 3α for an isotropic solid

In the formulas, L₀ is the starting length and V₀ the starting volume. ΔT is the temperature change (Δ, "delta", means "change in"). ΔL and ΔV are the changes in length and volume, and L and V are the final sizes. α (alpha) is the length rate and β (beta) the volume rate. Each is the share of its size an object grows per degree.

So twice the starting size, or twice the temperature change, gives twice the growth. An α of 12 × 10⁻⁶ /K means each meter grows by 12 micrometers for each kelvin of warming. A micrometer is a thousandth of a millimeter.

A solid expands in all three directions, so its volume rate is about three times its length rate. That holds for an isotropic solid, one that expands the same way in every direction. These simple formulas are accurate when the total change is a small part of the size. That's true for ordinary temperature ranges.

Worked example

A 30 m carbon-steel beam warms by 40 °C from a winter morning to a summer afternoon. How much longer does it get?

  1. α ≈ 12 × 10⁻⁶ /K = 0.000012 /K; ΔT = 40 °C = 40 K.
  2. ΔL = 0.000012 × 30 × 40 = 0.0144 m.
  3. In millimeters: 0.0144 × 1,000 = 14.4 mm.

Result: The beam grows by 14.4 mm, to 30.0144 m. That is a 0.048% change in length.

Understanding the result

A positive change is growth; a negative change is shrinking as it cools. The percentage change doesn't depend on the size. So a 3 m and a 300 m steel piece both grow by 0.048% for a 40 °C rise.

If the material can't move, the expansion turns into stress (push and pull inside it) instead. Steel held in place can build up very large forces. That's why rails buckle in heat waves, and why builders leave gaps or loops.

Assumptions and limitations

  • The expansion rate stays the same over the whole temperature range.
  • The material is free to expand and is not restrained.
  • The temperature is uniform through the object.
  • Preset rates are approximate. Real values depend on the alloy, mix, grade and temperature.
  • Water and many plastics have rates that change strongly with temperature. So large ranges need an average rate for that range.
  • Changes of state, such as freezing, aren't covered. Water expands by about 9% when it freezes, far more than this formula predicts.
  • Some materials, such as wood or fiber composites, expand differently in different directions (they're anisotropic).

Frequently asked questions

How big should an expansion gap be?

At least the growth (ΔL) for the biggest temperature swing the part will see, plus some extra room. For 30 m of steel and a 40 °C swing, that's 14.4 mm. For real installations, follow the building code that applies or the manufacturer's guidance.

Why do liquids expand more than solids?

Molecules in a liquid are held together more loosely. As they warm, they jiggle more, and that pushes them apart more easily. Ethanol, for example, expands roughly 30 times as much per degree as steel does by volume.

Does the container expand too?

Yes. A liquid in a tank overflows by the gap between the liquid's expansion and the tank's. With a steel tank and gasoline, the steel expands far less. So most of the liquid's expansion shows up as a rise in level.

What is invar?

It's a metal made of iron and nickel that barely grows when heated. Its rate is around 1.2 × 10⁻⁶ /K near room temperature. It's used where size must stay steady, such as in precision instruments.

How this calculator is checked

Method: Linear thermal expansion ΔL = αL₀ΔT and volumetric ΔV = βV₀ΔT (first-order approximations); β ≈ 3α for isotropic solids. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

Embed this calculator

Put this Thermal Expansion Calculator on your own website or blog for free. Paste this code where you want it to appear: