Volumetric thermal expansion:

          The volume of a solid also changes with the temperature change and is called volume thermal expansion or cubical thermal expansion.
          Consider a solid of the initial volume$ V_{0}$ at a certain temperature $ T_{0}$. On heating, the solid to a temperature T, let its volume becomes V, then
Change in the volume of a solid $ \triangle V = V - V_{0}$
And change in temperature $ \triangle T$ =$ T - T_{0}$

         Like linear expansion, the change in $ \triangle V$ is found to be proportional to its original volume $ V_{0}$ and change in temperature $ \triangle T $. Thus 
 $ \triangle V \alpha V_0 \triangle T $

or                                                              $ \triangle V $ = $ \beta V_0 \triangle T $ ……….  (i)
or                                                            $ V - V_{0} =  〖\beta V〗_{0} \triangle T$
or                                                               $V = V_{0}+\beta V_{0} \triangle T $
or                                                              $ V = V_{0} (1+ \beta \triangle T)$ ……….  (ii) 

          where \beta is the temperature coefficient of volume thermal expansion, using equation (i), we get 
         $ \beta $ = $ (\triangle V )/(V_{0} \triangle T )$                           (iii)

Coefficient of volume expansion $ \beta$:

           Thus, we can define the temperature coefficient of volume expansion $ \beta $ as the fractional change in its volume per kelvin change in temperature. The coefficient of linear expansion and volume expansion are related by the equation:
   $\beta = $3 \alpha$ ……………..  (iv)