Result

Thermal expansion

Δ L = α L_{0} Δ T, Δ V = β V_{0} Δ T

Statement

Linear and volumetric expansion, each proportional to the temperature change and the material's original size.

Symbols and units

$Δ L, Δ V$	change in length, volume	m, m³
$α, β$	linear, volumetric expansion coefficient	K⁻¹
$Δ T$	temperature change	K
$L_{0}, V_{0}$	original length, original volume	m, m³

When it applies

Small temperature changes, where the linear approximation to the true expansion curve holds.

Don't use when

Only a linear approximation valid for small temperature changes — over large temperature ranges the expansion coefficient itself varies and this approximation breaks down.

Derivation

To first order, expansion is proportional to ΔT and to the original size — the leading term of a more general (and generally more complex) length-versus-temperature relation.

Δ L \approx α L_{0} Δ T

Worked examples

Why β ≈ 3α

For a cube of side L, volume is L³; differentiating logarithmically:

\frac{d V}{V} = 3 \frac{d L}{L}

Since dV/V is exactly β\,dT and dL/L is exactly α\,dT, the factor of 3 follows directly — expansion happens along all three dimensions at once.

Watch out for

For an isotropic solid, β ≈ 3α — for a cube of side L, V = L³, so a small fractional length change dL/L produces a volume change dV/V = 3\,dL/L. Forgetting the factor of 3 is a common error.