### Theory:

Areal expansions:
When we heat a solid, if there is an increase in the area of the body, this is called Areal expansion. It is also known as Superficial expansion

The amount by which the area of a material increases when the temperature is raised one degree is called the coefficient of superficial expansion. The unit of coefficient of superficial expansion is ${K}^{-1}$.

The coefficient of superficial expansion can be designated by the symbol $\mathrm{\beta }$ (beta).

Areal expansion in solid

Consider a material with the area of ${A}_{1}$ at ${t}_{1}^{o}$ $$c$$. And, the material is heated upto ${t}_{2}^{o}$ $$c$$. Now, the length of the material is ${A}_{2}$.

Let $\mathrm{\beta }$ be the coefficient of linear expansion.

We know that,

The change in area $$ΔA$$ is proportional to the original area ${A}_{1}$, rise in temperature $$ΔT$$, and material type.

$\begin{array}{l}\mathrm{\Delta }A\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{A}_{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{A}_{1}\phantom{\rule{0.147em}{0ex}}\mathit{and}\\ \\ \mathrm{\Delta }T\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{t}_{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{t}_{1}\\ \\ \mathrm{\Delta }A\phantom{\rule{0.147em}{0ex}}\propto \phantom{\rule{0.147em}{0ex}}{A}_{1}\mathrm{\Delta }T\\ \\ \mathrm{\Delta }A\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }{A}_{1}\mathrm{\Delta }T\end{array}$

Rearranging the above equation,

$\begin{array}{l}\mathrm{\beta } =\phantom{\rule{0.147em}{0ex}}\frac{{A}_{2}-{A}_{1}}{{A}_{1}\left({t}_{2}-{t}_{1}\right)}\\ \mathit{Where},\\ {A}_{1}\mathit{and}\phantom{\rule{0.147em}{0ex}}{A}_{2}\phantom{\rule{0.147em}{0ex}}\mathit{are}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{areas}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{material}\phantom{\rule{0.147em}{0ex}}\mathit{before}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{after}\phantom{\rule{0.147em}{0ex}}\mathit{heating}\phantom{\rule{0.147em}{0ex}}\mathit{respectively}.\\ {t}_{1}\mathit{and}\phantom{\rule{0.147em}{0ex}}{t}_{2}\phantom{\rule{0.147em}{0ex}}\mathit{are}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{temperatures}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{material}\phantom{\rule{0.147em}{0ex}}\mathit{before}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{after}\phantom{\rule{0.147em}{0ex}}\mathit{heating}\phantom{\rule{0.147em}{0ex}}\mathit{respectively}.\\ \mathrm{\beta }\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{Areal}\phantom{\rule{0.147em}{0ex}}\mathit{coefficient}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{expansion}\phantom{\rule{0.147em}{0ex}}\end{array}$

The $\mathrm{\beta }$ will vary for different materials.