Dear May,

you may simply calculate the volumetric expansion coefficient via the increase of the volume delta V when expanding in all three dimensions.

For simplicity take a cube expanding its edges from Lo to L by delta L (dL):

L= Lo+dL= Lo*(1+dL/Lo) = Lo*(1+alpha); with alpha = linear expansion coefficient dL/Lo.

For the cube volume we start with Vo =Lo³ and V=L³= [Lo*(1+alpha)]³ and let V =Vo*(1+gamma) be the goal of calulations (gamma= volumetric expansion coefficient).

So we will have:

V=Lo³*(1+alpha)³= Vo*(1+alpha)³= Vo*(1+3alpha+3alpha²+alpha³).

However, alpha in reality is a very small number; so its square and even its cube can be neglected relative to alpha.

So we will have: V=Lo³*(1+alpha)³= Vo*(1+alpha)³~ Vo*(1+3alpha).

Thus the volume expansion coeficient gamma willl be: gamma = 3 alpha.

That is clear, Thank you :)