Relation between coefficient of linear expansion and coefficient of real expansion:
Consider a thin rectangular parallelopiped solid of length, breadth, height and volume $$l_o\ b_o\ h_o\ V_o$$ at temperature $$0^{o}C$$. Let the solid be heated at some higher temperature say to C.
Let $$l,b,h,V$$ be the length, breadth, height and volume at temperature to C.
Then the original volume $$V_o=l_ob_oh_o$$ ....(1)
Consider linear expansion
length $$l=l_o(1+\alpha t)$$
breadth $$b=b_o(1+\alpha t)$$
height $$h=h_o(1+\alpha t)$$
where $$\alpha =coefficient \ of \ linear\ expansion$$
Final volume $$V=lbh=l_o(1+ \alpha t)×b_o(1+ \alpha t)×h_o(1+\alpha t)$$
$$V=l_ob_oh_o(1+3 \alpha t+3 \alpha^2t^2+ \alpha^3t^3)$$
\alpha
Now, $$ \alpha $$ is very small hence $$ \alpha^2$$ is still small, hence quantity $$ \alpha^2t^2\ \alpha^3t^3$$ can be neglected.
$$V=V_o(1+3\alpha t)$$ .....(2)
Considering cubical expansion of the solid
$$V=V_o(1+\gamma t)$$
From (1) and (2),
$$\gamma=3\alpha$$