Last updated: 2026-06-15

Thermal Expansion

Heat a solid and it grows; cool it and it shrinks. The effect is tiny for everyday objects but impossible to ignore in engineering, where a heated bridge, rail or pipe can move by centimetres. This calculator uses the linear expansion formula to turn an original length, a material's expansion coefficient and a temperature change into the change in length and the new length.

Where the formula comes from

Inside a solid, atoms vibrate about fixed positions. Heating them makes the vibrations larger, and because the forces between atoms are not perfectly symmetric, the average spacing grows slightly. Each bit of the material stretches by the same fraction per degree, so the total stretch is proportional both to the original length and to the temperature change. The constant of proportionality is the coefficient of linear thermal expansion, $\alpha$.

Formula

Because only the temperature difference appears, you get the same answer in kelvin or in degrees Celsius. A negative $\Delta T$ (cooling) gives a negative $\Delta L$ — the object contracts.

Worked example

A 1 m steel bar ($\alpha = 12\ \text{ppm/K} = 12\times10^{-6}\ \text{K}^{-1}$) heated from $20\ ^\circ\text{C}$ to $100\ ^\circ\text{C}$:

Under a millimetre over a metre — small, but a 100 m span would move nearly a hundred times as much.

Area and volume

The same coefficient governs how area and volume grow, just scaled up. Area expands at about twice the linear rate and volume at three times, so for a solid that expands equally in all directions $\Delta A \approx 2\alpha\,A_0\,\Delta T$ and $\Delta V \approx 3\alpha\,V_0\,\Delta T$. This is why a metal lid expands faster than the glass jar it sits on, loosening when run under hot water.

Limitations

The formula assumes $\alpha$ is constant over the temperature range, which holds well for ordinary spans but drifts at very high or very low temperatures. It also assumes the material is free to expand; if it is clamped, it cannot grow and instead builds up large thermal stresses, which is why unrelieved expansion can buckle rails or crack masonry. Some materials, like water near 0 °C, even contract on warming over a small range — behaviour this linear model does not capture.