Mar 18, 2017

Linear expansion $\alpha$ is the length increase when an object is heated.
Volumetric expansion $\gamma$ is the volume increase.

#### Explanation:

If a block of material is heated, it will expand in all directions.

So in a block of $1 m \times 1 m \times 1 m$ with a linear coefficient of $\alpha$ the volume increases from $1 {m}^{3}$ to ${\left(1 + \alpha\right)}^{3} {m}^{3}$ per degree.

Since $\alpha$ is very small (in the region of ${10}^{- 5} / K$), we can say that ${\left(1 + \alpha\right)}^{3} \approx 1 + 3 \alpha$ with sufficient precision.

So the co-efficient of volumetric expansion: $\gamma \approx 3 \alpha$

Note:
${\left(1 + \alpha\right)}^{3} = 1 + 3 \alpha + 3 {\alpha}^{2} + {\alpha}^{3}$ (Pascal)
Since $\alpha$ is very small ($\approx {10}^{-} 5$), ${\alpha}^{2}$ will be in the region of ${10}^{- 10}$ and ${\alpha}^{3} \approx {10}^{- 15}$.
These can be neglected, hece ${\left(1 + \alpha\right)}^{3} \approx 1 + 3 \alpha$

Note 2:
Of course, with liquids you can only measure volume expansion.