Thermal Expansion Calculator

Heating a solid makes it grow in proportion to its original size and to the temperature change:

$\Delta L = \alpha L_0 \Delta T$

Symbols and units:

• $\Delta L$ — change in length, metres (m)
• $\alpha$ — coefficient of linear expansion, per kelvin (K⁻¹, numerically the same as °C⁻¹)
• $L_0$ — original length, metres (m)
• $\Delta T$ — temperature change, kelvin or °C (a difference, so the two scales agree)

The new length is $L = L_0(1 + \alpha\Delta T)$.

When it applies: to solids over moderate temperature ranges, where $\alpha$ is effectively constant.

The assumption people forget: $\alpha$ itself varies with temperature. Over a few hundred degrees the constant-$\alpha$ form is fine; across a phase change or a very wide range it is not, and you need a temperature-dependent $\alpha$ or tabulated expansion data.

For an isotropic material the same $\alpha$ drives all three:

$\Delta A = 2\alpha A_0 \Delta T \qquad \Delta V = \beta V_0 \Delta T,\ \ \beta \approx 3\alpha$

$A$ in m², $V$ in m³, $\beta$ in K⁻¹.

A hole expands too. A hole in a heated plate gets larger, not smaller — the material around it grows outward, so the hole scales exactly as if it were made of the same metal.

The assumption people forget: $\beta = 3\alpha$ holds only for isotropic solids. Anisotropic crystals and composites expand differently along different axes, and liquids need their own measured $\beta$.

• Converting °C to K for $\Delta T$ — a change of $35$ °C is a change of $35$ K. Adding $273.15$ to a temperature difference is wrong.
• Using $\alpha$ where $\beta$ belongs — volume expansion is roughly three times linear expansion.
• Mixing length units — if $L_0$ is in metres, $\Delta L$ comes out in metres; multiply by $1000$ for millimetres.
• Dropping the $10^{-6}$ — coefficients are quoted in units of $10^{-6}$ K⁻¹, and forgetting it inflates the answer a million-fold.
• Assuming a hole shrinks — it expands with the plate.
• Treating expansion joints as optional — a restrained member that cannot expand develops thermal stress $\sigma = E\alpha\Delta T$ instead, which is what cracks concrete and buckles rails.
• Forgetting that both parts of an assembly move — a steel bolt in an aluminium housing loosens on heating because the aluminium grows nearly twice as fast. What governs the clearance is the difference in $\alpha$ between the two materials, not either value on its own.