text: For isotropic materials the volumetric thermal expansion coefficient is three times the linear coefficient:
  This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, for small differential changes, one-third of the volumetric expansion is in a single axis. As an example, take a cube of steel that has sides of length . The original volume will be  and the new volume, after a temperature increase, will be
  We can easily ignore the terms as ΔL is a small quantity which on squaring gets much smaller and on cubing gets smaller still.
  So
  The above approximation holds for small temperature and dimensional changes (that is, when  and  are small), but it does not hold if trying to go back and forth between volumetric and linear coefficients using larger values of . In this case, the third term (and sometimes even the fourth term) in the expression above must be taken into account.
  Similarly, the area thermal expansion coefficient is two times the linear coefficient:
  This ratio can be found in a way similar to that in the linear example above, noting that the area of a face on the cube is just . Also, the same considerations must be made when dealing with large values of .
  Put more simply, if the length of a cubic solid expands from 1.00 m to 1.01 m, then the area of one of its sides expands from 1.00 m to 1.02 m and its volume expands from 1.00 m to 1.03 m.